# Correlated default models driven by a multivariate regime-switching shot noise process

Correlated default models driven by a multivariate regime-switching shot noise process Abstract We develop a reduced-form credit risk model with regime-switching intensities to investigate the pricing of a credit default swap (CDS) contract. We assume that the defaults of all the names are driven by some shock events. The arrivals of the shock events and the interest rate are modelled by a multivariate regime-switching shot noise process. We provide the flexibility that the model parameters, including the intensities and the jump sizes of the jump component, can switch over time according to a continuous-time, finite-state Markov chain. The states of the chain may be interpreted as different states of an economy or different stages of a business cycle. Based on the joint Laplace transform of the regime-switching shot noise processes, we derive the explicit formulas for the spreads of CDS contract with and without counterparty risk. Numerical results illustrate changes of market regimes have a significant effect on the spread. 1. Introduction In recent years credit derivatives, which are derivatives with a pay-off linked to the credit loss in a portfolio, enjoyed one of the largest growth of all the markets. Credit default swaps (CDS) are the most widely traded form of credit derivatives. A CDS is a financial swap agreement between the buyer of the default protection on a reference risky entity and the seller of the default protection. The protection seller receives fixed periodic payments (CDS spread) from the protection buyer, in return for compensating the buyer’s losses on the reference entity when a credit event occurs. This article focuses on valuing a single-name CDS with and without counterparty risk. There are two primary approaches for pricing credit derivatives, the structural approach and the reduced-form approach. The structural approach, pioneered by Black & Scholes (1973) and Merton (1974), uses the evolution of a firm’s structural variable to determine the time of default, whereas the reduced-form approach, introduced by Jarrow & Turnbull (1995), Duffie & Singleton (1999), treats default as a jump process with an exogenous intensity. Comparing with the structural models, the reduced-form models are more flexible and tractable in dealing with the pricing of multi-name credit derivatives. The works on the reduced-form models differ mainly in specifying default intensities of individual entities and their default dependence. There exist four major approaches to introduce default correlation within the reduced-form framework: the conditionally independent approach, the copula approach, the default contagion models and the common shock models. In the conditionally independent default models, one may set the default intensities of the firms in the portfolio be driven by a common set of macro-economic factors. Therefore, conditional on the realization of the macro-economic state variables, the default times are mutually independent. Some works on the conditionally independent default models include Duffie & Gârleanu (2001), Di Graziano & Rogers (2009), Ge et al. (2015). In the copula models, the dependence structure is linked through a copula function (see, Schönbucher & Schubert, 2001 and Hull & White, 2004). Default contagion is another approach to model the default correlation. The contagion models study the direct interaction of firms in which the default probability of one firm may change upon defaults of some other firms in the portfolio (see Davis & Lo, 2001 and Dong et al., 2016). The common shock models are based on the idea that a firm’s default is driven by exogenous events, for example, policy events, natural catastrophes events, etc. Therefore, simultaneous defaults may occur under the common shock models. Some works on common shock models include Lindskog & McNeil (2003), Giesecke (2003) and Bielecki et al. (2012). This article focuses on the common shock models. In an earlier common shock model, a firm’s default is driven by some idiosyncratic or economy-wide shocks, whose arrivals are modelled by some mutually independent Poisson processes (see, e.g. Lindskog & McNeil, 2003 and Giesecke, 2003). In recent years, there have been numerous contributions on how to model the arrival processes of the shock events by stochastic processes. See, for example, Bielecki et al. (2012) introduced a common shock Markov copula model, in which the arrival intensities are specified in the form of some Cox-Ingersoll-Ross (CIR) model processes. Bielecki et al. (2013) investigated the calibration methodology under a common shock Markov copula model, where the stochastic intensities are specified in the form of an extended CIR process. However, none of the above-mentioned works incorporated jumps into the modelling of the arrival intensities. Shot noise processes, a class of pure jump processes, are particularly useful in the arrival processes since they allow for explicit solutions of many important quantities in derivative pricing. See, for example, Gaspar & Schmidt (2010) consider a multivariate default model driven by shot noise processes and show that a shot noise process gives a superior to historical data as well as a better fit in calibration comparing with an affine jump-diffusion model proposed in Duffie & Gârleanu (2001). The shot noise process (see also, e.g. Cox & Isham, 1986 and Dassios & Jang, 2003) can be well used to measure the impact on the intensities of major events. As was pointed out in Dassios & Jang (2003), the shot noise process measures the frequency, magnitude and time period needed to go back to the previous level of intensity immediately after major events occur. However, empirical studies point to the existence of different regimes in the default risk valuation (see, e.g. Davies, 2004 and Giesecke et al., 2011). Intuitively, the default intensities should be related with the macro-economic conditions since we have witnessed that the recent global financial crisis has had a significant impact on the international financial markets, in particular on the values of credit derivatives. In fact, default risk is much influenced by the business cycles or macro-economy. Default risk typically declines during economic expansion because strong earnings keep overall defaults rates low. Default risk increases during economic recession because earnings deteriorate, making it more difficult to repay loans or make bond payments. Credit derivatives are long term instruments and thus it is very important to develop more appropriate models for valuation and risk management of credit products, which can take into account changes of market regimes due to the crisis. Regime-switching models were introduced by Hamilton (1989) to financial econometrics and economists. One of the main features of these models is that model dynamics are allowed to change over time according to the state of an underlying Markov chain. Regime switches are often interpreted as structural changes in macro-economic conditions and in different stages of business cycles. Recently, regime-switching models have gained immense popularity in the credit risk modelling (see, e.g. Giesecke et al., 2011; Bo et al., 2013; Dong et al., 2014a, 2016). Therefore, motivated by Gaspar & Schmidt (2010), Giesecke et al. (2011), Lindskog & McNeil (2003), we consider a common shock model, in which the arrival intensities of the shock events and the interest rate are governed by a multivariate regime-switching shot noise process. Note that, (Dong et al., 2014a, 2016) also investigate the application of a multivariate regime-switching shot noise process in the credit portfolio model. However, the regime-switching shot noise process considered in Dong et al. (2014a, 2016) is Markovian. More precisely, it is a mean-reverting Markov process or a so-called Ornstein–Uhlenbeck process. By using the Markov property, Dong et al. (2014a, 2016) derive the Laplace transform of the regime-switching shot-noise process via a martingale method. Typically, the regime-switching shot-noise processes are not Markovian. However, in more general cases one looses Markovianity, the martingale method adopted by Dong et al. (2014a, 2016) cannot be applied to derive the Laplace transform of the regime-switching shot noise process. In this article, we consider a more general regime-switching shot-noise process which looses Markovianity and provide an approach to derive the Laplace transform of the regime-switching shot noise process. Summing up, this article aims at providing a flexible and tractable model for correlated defaults which can incorporate the impacts of the changes of market regimes on the interest rate and the default intensities. The contribution of this article is to provide a reduced-form credit risk model with regime-switching shot noise intensities which allows for analytic formulas for the CDS spreads with and without counterparty risk. The structure of the article is organized as follows: in Section 2, we provide the basic setup of our framework, where a common shock model with regime-switching shot noise intensities is introduced and some preliminary results are presented. Section 3 derives the joint Laplace transform of the shot noise processes and the joint survival distributions. In Section 4, we give the closed-form formulas for the CDS spreads with and without counterparty risk. Section 5 performs some numerical experiments to examine how the regime-switching shot noise intensities affect the CDS spread. Section 6 concludes. The proofs are presented in the appendix. 2. Default dependence and some preliminary results In this section, we shall consider a common shocks portfolio credit risk model. Consider a continuous-time model with a finite time horizon $$\mathcal{T}=[0, T]$$ with $$T < \infty$$. Let $$\{\Omega,\Im,\{\Im_t \}_{0\leq t\leq T}, P\}$$ be a filtered complete probability space, where $$P$$ is the risk neutral measure and $$\{\Im_t \}_{0\leq t\leq T}$$ is a filtration satisfying the usual conditions. Throughout the article, it is assumed that all random variables are well defined on this probability space and $$\Im_T-$$measurable. Denote by D(t,T)=e−∫tTrsds the stochastic discount factor at time $$t$$ for maturity $$T,$$ where the stochastic interest rate $$r_t$$ will be specified later. Consider a CDS contract with notional value one, continuous spread rate payments and maturity $$T.$$ Indices 0, 1 and 2 refer to quantities related to the investor, the reference entity and the counterparty. Denote by $$\tau_{0}, \tau_1$$ and $$\tau_{2}$$ be the default times of the investor, the reference entity and the counterparty, respectively; and denote by $$R_{1}$$ the recovery of the reference entity which is supposed to be constant. In this article, we assume that all the cash flows and prices are considered from the perspective of the investor. Now we construct a default dependence structure within the framework of a common shock model. Following Lindskog & McNeil (2003), we assume there are $$m$$ event types called factor names, which can generate events potentially causing joint defaults. Assume that there exists a process $$X:=\{X_t\}_{t\geq 0}$$ which drives all of the arrivals of the $$m$$ types of shock events and the interest rate. Let $$X$$ be a homogenous continuous-time, finite-state, irreducible Markov chain with generator $$Q=(q_{ij})_{i,j=1,2,\cdots,N},$$ generating a filtration $$\Im^X:=\{\Im_t^X|t\in\mathcal{T}\}.$$ The states of the Markov chain $$X$$ are interpreted as different states of a macro-economy or different stages of a business cycle. Without loss of generality, we identify the states of the Markov chain $$X$$ with a set of standard unit vectors $$\mathcal{E}=\{e_1, e_2, \cdots, e_N\},$$ where $$e_i = (0,\cdots, 0, 1, 0,\cdots, 0)^{*}\in R^N,$$$$*$$ denotes the transpose of a vector or a matrix. Elliott et al. (1994) provide the following semi-martingale decomposition for $$X$$: dXt=Q∗Xtdt+dMt, (2.1) where $$\mathcal{M}_{t}$$ is an $$R^N$$-valued martingale with respect to the filtration $$\Im^X$$ under the measure $$P.$$ Assume that the $$m$$ types of shock events arrive as Cox processes $$\{N^{1}(t),t\geq 0\},\cdots,\{N^{m}(t),t\geq 0\}$$ with stochastic intensities $$\lambda^{1}_{t},\cdots, \lambda^{m}_{t}$$ generating a filtration $$\Im^{\boldsymbol{\lambda}}:=\{\Im_t^{\boldsymbol{\lambda}}|t\in\mathcal{T}\},$$ where $$\Im_t^{\boldsymbol{\lambda}}=\Im_t^{\lambda^1}\vee \Im_t^{\lambda^2}\vee\cdots\vee \Im_t^{\lambda^m},$$ with $$\Im_t^{\lambda^k}=\sigma(\lambda_s^k,0\leq s\leq t).$$ Furthermore, assume given $$\Im_t^{\boldsymbol{\lambda}},$$$$\{N^{1}(t),t\geq 0\},\cdots,\{N^{m}(t),t\geq 0\}$$ are mutually independent. Let $$\langle.,.\rangle$$ denote a scalar product in $$R^{N},$$ that is, for any $${\bf x}, {\bf y}\in R^{N},$$$$\langle{\bf x}, {\bf y}\rangle=\sum\limits_{i=1}^{N}x_{i}y_{i}.$$ In order to model the interest rate and the arrival intensities of the shock events, we first introduce the regime-switching shot noise process. Let $$M(t)$$ be a point process with jump times $$(T_n)_{n\geq 1}.$$ If $$X_s=e_i$$ for all $$s$$ in a small interval $$(t,t+h],$$ then $$M(t+h)-M(t)$$ has a Poisson distribution with parameter $$\lambda_i>0.$$ We further assume that given the Markov chain $$X,$$ the process $$\{M(t)\}_{t\geq 0}$$ has independent increments. Then P(M(t+h)=n+1|M(t)=n,Xs=ei,t<s≤t+h)=λih+o(h). The process $$\{M(t)\}_{t\geq 0}$$ is called a regime-switching Poisson process with intensity $$\lambda_t=\langle \boldsymbol{\lambda},X_t\rangle$$ for a constant positive vector $$\boldsymbol{\lambda}=(\overline{\lambda}_1,\cdots,\overline{\lambda}_N)^*\in R^N$$, which is a special case of Cox processes. Let $$Y_1,Y_2,\ldots$$ be a sequence of $$R^d$$-valued random variables. Given $$X_s=e_i,$$ the jump amounts $$Y_1,Y_2,\ldots$$ are independent and identically distributed with distribution function $$F_i(.)$$ and they are independent of $$M(t).$$ The process Jt=∑k=1M(t)Yk is called a regime-switching compound Poisson process, which is widely used in finance and insurance domain (see, e.g. Lu & Li, 2009 and Zhang et al., 2012). Consider a measurable function $$h:R^d\times R\rightarrow R$$ and define the following process by St=S0+∑Tn≤th(Yn,t−Tn)≐S0+Lt, (2.2) where $$S_0=\langle {\bf S}_0, X_0\rangle$$ for a vector $${\bf S}_0=(\overline{S}_1, \overline{S}_2,\cdots, \overline{S}_N)^*\in R^N.$$ Then $$S:=\{S_t\}_{t\geq 0}$$ is a regime-switching shot noise process starting from $$S_0$$ driven by a regime-switching compound Poisson process. If there is no regime switching and $$S_0=0$$, then the model (2.2) is the same as the shot noise process driven by a compound Poisson process, which is investigated in Gaspar & Schmidt (2010). Note that, a regime-switching shot-noise process driven by a regime-switching compound Poisson process can be extended to a regime-switching shot noise process driven by a marked point process with regime switching, which will be considered in our future’s research. Integrating $$S_u$$ from $$0$$ to $$t$$ yields ∫0tSudu=S0t+∫0t∑Tn≤uh(Yn,u−Tn)du=S0t+∑Tn≤tH(Yn,t−Tn), (2.3) where $$H(y,u)=\int_0^uh(y,v)dv.$$ Remark 2.1 As was pointed out in Gaspar & Schmidt (2010), if used for intensity modelling or short interest rate modelling, $$S$$ has to be non-negative. Therefore, we require $$Y_n\geq 0$$ as well as $$h>0.$$ For the choice of $$h,$$ one usually works with $$h$$ of multiplicative type, such that $$h(u,t)=ug(t)$$ for some function $$g$$. In this article, we also consider the case that $$h$$ is of the form of multiplicative type. In particular, (a) if $$g(t)=1,$$ then the process $$L_t$$ in (2.2) becomes a regime-switching compound Poisson process; (b) if $$g(t)=e^{-at},$$ where $$a>0$$ is a constant, then $$L_t$$ is a mean-reverting regime-switching Markov process, and it solves the stochastic differential equation dLt=−aLtdt+dJt. (2.4) As was explained by Dassios & Jang (2003), the process $$L_t$$ in (2.4) measures the frequency, magnitude and time period needed to go back to the previous level of intensity immediately after shock events occur. The process $$L_t$$ decreases until another shock event occurs, which will result in a positive jump in the regime-switching Markov process. Dong et al. (2014a, 2016) considered the pricing of a CDS contract under the model (2.4) via a martingale method. This article will investigate a general regime-switching shot noise process (2.2) by using a different approach. Now, we model the interest rate and the arrival intensities by a multivariate regime-switching shot noise process driven by a multivariate regime-switching compound Poisson process. Assume $$M_0(t),$$$$M_1(t)$$, $$\ldots, M_m(t)$$ are $$m+1$$ conditionally independent regime-switching Poisson processes. For each $$k=0,1,\ldots,m,$$ the intensity of $$M_k(t)$$ is given by $$\mu_k(t)=\langle \boldsymbol{\mu}_k, X_t\rangle,$$ for a positive vector $$\boldsymbol{\mu}_k=(\mu_{k}^{1},\ldots,\mu_{k}^N)^*\in R^N,$$ and denote by $$(T_n^k)_{n\geq 1}$$ the jump times of $$M_k(t).$$ Given $$X_t,$$ let the non-negative jump amounts $$\{Y_1^0, Y_2^0,\ldots\}, \{Y_1^1, Y_2^2,\ldots\},\ldots, \{Y_1^m, Y_2^m,\ldots\}$$ be mutually independent and independent of $$M_k(t),k=0,1,\ldots,m.$$ Assume given $$X_t=e_i,$$$$\{Y_1^k, Y_2^k,\ldots\}$$ is a sequence of independent and identically distributed $$R^+$$-valued random variables with distribution function $$F^{ki}(.)$$ satisfying $$\int_0^{\infty}yF^{ki}(dy)<\infty$$ for each $$k=0,1,\ldots,m,i=1,2,\ldots, N.$$ That is to say, the random variables $$Y_1^k, Y_2^k,\ldots$$ have a conditional distribution $$F_t^k(.)=\langle {\bf F}^k(.),X_t\rangle,$$ where $${\bf F}^k(.)=(F^{k1}(.),\ldots, F^{kN}(.))^*\in R^N.$$ Then the interest rate and the arrival intensities are modelled by rt=r0+∑Tn0≤tYn0h0(t−Tn0), (2.5) and λtk=λ0k+∑Tnk≤tYnkhk(t−Tnk)+∑Tn0≤tYnkhk(t−Tn0),k=1,2…,m. (2.6) Here, $$r_{0}=\langle {\bf r},X_{0}\rangle, \lambda_{0}^k=\langle\boldsymbol{\lambda}^k,X_{0}\rangle,$$ where $${\bf r}=(\overline{r}_{1},\overline{r}_{2},\ldots,\overline{r}_{N})^{*}\in R^N, \boldsymbol{\lambda}^k=(\overline{\lambda}_{1}^k,\overline{\lambda}_{2}^k,\ldots,\overline{\lambda}_{N}^k)^{*}\in R^N$$ with $$\overline{r}_i>0, \overline{\lambda}_i^k>0$$ for each $$k=1,\ldots,m, i=1,2,\ldots,N;$$$$h_l(.)$$ is an $$R^+$$-valued deterministic function satisfying $$\int_0^{t}h_l(s)ds<\infty, \forall t<\infty,$$ for each $$l=0,1,\ldots,m$$. Remark 2.2 Let $$\hat{M}_k(t)=M_k(t)+M_0(t).$$ Then the counting process $$\hat{M}_k(t)$$ is still a regime-switching Poisson process with intensity given by $$\langle \boldsymbol{\mu}_k+\boldsymbol{\mu}_0,X_t\rangle,$$ for each $$k=1,2,\ldots,m.$$ If we denote by $$(\hat{T}_n^k)_{n\geq 1}$$ the jump times of $$\hat{M}_k(t),$$ then (2.6) can be expressed as λtk=λ0k+∑T^nk≤tYnkhk(t−T^nk),k=1,2…,m. Note that, the stochastic interest rate and the intensities $$\lambda^{1}_{t},\ldots, \lambda^{m}_{t}$$ given by (2.5)–(2.6) are driven by a multivariate regime-switching shot noise process with common jumps. As was explained in Siu (2010), the interest rate jumps because of some extraordinary market events, such as market crashes, interventions by central banks or monetary authorities. Therefore, the regime-switching shot noise process can well describe the impact on the interest rate of the major market events. Since extraordinary market events which trigger jumps in the interest rate may also trigger simultaneous jumps in the arrival intensities of the shock events, we assume the intensities are all impacted by $$M_0(t)$$ which counts the number of the arrivals of extraordinary market events. The intensities are still impacted by some factors specified to each group. More precisely, $$M_k(t)$$ only counts the number of the factor events specified to group $$k.$$ Integrating $$r_u$$ from $$0$$ to $$t$$ yields ∫0trudu=r0t+∫0t∑Tn0≤uh0(u−Tn0)Yn0du=r0t+∑Tn0≤tH0(t−Tn0)Yn0, (2.7) where $$H_0(u)=\int_0^uh_0(v)dv.$$ Similarly, ∫0tλukdu=λ0kt+∑Tn0≤tHk(t−Tn0)Ynk+∑Tnk≤tHk(t−Tnk)Ynk,k=1,2,…,m, (2.8) where $$H_k(u)=\int_0^uh_k(v)dv.$$ Now, we construct a default dependence structure by thinning the Cox processes $$N^{e}(t),e=1,\ldots,m.$$ Following Lindskog & McNeil (2003), let $$\{N_j(t), t \geq 0\}$$ be a counting process that counts shocks in the interval $$(0, t]$$ resulting in default of name $$j$$ in $$(0, t], j=0,1,2.$$ At the $$k$$th occurrence of an event of type $$e$$ the Bernoulli variable $$I^{(e)}_{j,k}$$ indicates whether an event of name $$j$$ occurs,$$e=1,2,\ldots,m,j=0,1,2.$$ For simplicity, the vectors Ik(e)=(I0,k(e),I1,k(e),I2,k(e))∗ for $$k = 1, \ldots, N^{e}(t)$$ are considered to be independent and identically distributed with a multivariate Bernoulli distribution. Furthermore, we assume, for each $$k = 1, \ldots, N^{e}(t),$$ the random variables $$I_{0,k}^{(e)},I_{1,k}^{(e)},I_{2,k}^{(e)}$$ are mutually independent. That is, P(Ij1,k(e)=ij1,Ij2,k(e)=ij2)=P(Ij1,k(e)=ij1)P(Ij2,k(e)=ij2),j1,j2∈{0,1,2},ij1,ij2∈{0,1}, and P(I0,k(e)=i0,I1,k(e)=i1,I2,k(e)=i2)=P(I0,k(e)=i0)P(I1,k(e)=i1)P(I2,k(e)=i2),i0,i1,i2∈{0,1}. This article only considers the case that the probability $$P(I_{j,k}^{(e)}=1)$$ is a constant and write for $$P(I_{j,k}^{(e)}=1)=p_{ej},$$ and $$P(I_{j,k}^{(e)}=0)=\overline{p}_{ej},$$ where $$\overline{p}_{ej}=1-p_{ej}.$$ Denote by $$\mathcal{S}$$ the collection of all the non-empty subsets of the set $$\{0, 1, 2\}.$$ For notational convenience, throughout this article, let λt=(λt1,…,λtm)∗∈Rm,Pj=(p1j,…,pmj)∗∈Rm,j=0,1,2,Ps=(∏j∈sp1j∏l∉sp¯1l,…,∏j∈spmj∏l∉sp¯ml)∗∈Rm,s∈S,Plj=(1−p¯1lp¯1j,…,1−p¯mlp¯mj)∗∈Rm.P=(1−∏j=02p¯1j,…,1−∏j=02p¯mj)∗∈Rm. From the above assumptions, the counting process $$N_j(t)$$ can be expressed as Nj(t)=∑e=1m∑k=1Ne(t)Ij,k(e):=∑e=1mNj(e)(t), where $$N^{(e)}_j(t)=\sum\limits_{k=1}^{N^{e}(t)}I_{j,r}^{(e)}.$$ Obviously, given $$\Im_t^{\boldsymbol{\lambda}},$$ the processes $$N^{(e)}_j(t), e=1,\ldots, m$$ are independent non-homogeneous Poisson processes with intensities $$\lambda_t^{e}p_{ej}, e=1,\ldots, m,$$ and the counting process $$N_j(t)$$ is a non-homogeneous Poisson processes with intensity $$q_j(t)=\langle \boldsymbol{\lambda}_t, {\bf P}^j\rangle.$$ By using the process $$N_j(t),$$ we define the default time of name $$j$$ as τj=inf{t≥0:Nj(t)=1},j=0,1,2. So, we have P(τj>t|ℑtλ)=P(Nj(t)=0|ℑtλ)=e−∫0tqj(u)du, and P(τj>t)=P(Nj(t)=0)=E[e−∫0tqj(u)du]. Define the default process as Htj=1{τj≤t},j=0,1,2. Denote the filtration by ℑt=ℑtX∨ℑtr∨ℑtλ∨ℑt0∨ℑt1∨ℑt2, where $$\Im_t^r=\sigma(r_s:0\leq s\leq t),$$ and $$\Im_{t}^{j}=\sigma(H_{u}^{j}:0\leq u\leq t), j=0,1,2.$$ In order to derive the joint survival distribution, we introduce some auxiliary counting processes. For each $$s\in \mathcal{S},$$ define a counting processes $$N_s(t),$$ which counts shocks in the interval $$(0, t]$$ resulting in defaults of all the names in s only. Thus if $$s=\{1,2\},$$ then $$N_s(t)$$ counts shocks which cause simultaneous defaults of names 1 and 2, but not of name 0. Similar to Lindskog & McNeil (2003), we have Ns(t)=∑e=1m∑j=1N(e)(t)∑s′:s′⊇s(−1)|s′|−|s|∏k∈s′Ik,j(e), where $$|s|,s\in \mathcal{S}$$ is the cardinality of the set $$s,$$$$\sum\limits_{s':s'\supseteq s}(-1)^{|s'|-|s|}\prod\limits_{k\in s'}I_{k,j}^{(e)}$$ is an indicator random variable which takes the value 1 if the $$j$$th shock of type $$e$$ causes defaults of all type in $$s$$ only, and the value 0 otherwise. Lindskog & McNeil (2003) proved that when $$N^e(t), e=1,2,\ldots,m$$ are mutually independent Poisson processes, the process $$N_s(t)$$ is a Poisson process and $$\{N_s(t), t \geq 0\}$$ for $$s \in\mathcal{S}$$ are independent Poisson processes. By the same augments as in Lindskog & McNeil (2003), we can conclude that in our model, $$\{N_s(t), t \geq 0\},$$ for $$s \in\mathcal{S},$$ are conditionally independent Cox processes, and for each $$s\in\mathcal{S},$$ the process $$N_s(t)$$ is a Cox process with intensity qs(t)=∑e=1mλte∑s′:s′⊇s(−1)|s′|−|s|∏j∈s′pej. It is easy to check that q{j}(t)=⟨λt,P{j}⟩,{j}∈S,q{l,j}(t)=⟨λt,P{l,j}⟩,{l,j}∈S,q{0,1,2}(t)=⟨λt,P{0,1,2}⟩, and ∑s:s∈Sqs(t)=⟨λt,P⟩. Clearly, we have $$N_j(t)=\sum\limits_{s\in \mathcal{S}:j\in s}N_s(t).$$ In order to derive the joint survival distribution, define the first jump time of $$N_s(t)$$ as τs=inf{t≥0:Ns(t)=1},s∈S. Since $$\{N_s(t), t \geq 0\},$$ for $$s \in\mathcal{S},$$ are conditionally independent Cox processes, the stopping times $$\tau_s,$$ for $$s\in\mathcal{S},$$ are also conditionally independent. Furthermore, the conditional and unconditional survival distributions for $$\tau_s$$ can be expressed as P(τs>t|ℑtλ)=P(Ns(t)=0|ℑtλ)=e−∫0tqs(u)du and P(τs>t)=P(Ns(t)=0)=E[e−∫0tqs(u)du]. Then by using the definition of $$\tau_s,$$ the default time $$\tau_j$$ can be redefined as τj=inf{t≥0:∑s∈S:j∈sNs(t)=1}=mins∈S:j∈sτs,j=0,1,2. (2.9) From (2.9), we can derive the conditional joint survival distribution presented in the following proposition. results. Proposition 2.1 For $$t_j\geq 0, j=0,1,2,$$ we have P(τ0>t0,τ1>t1,τ2>t2|ℑt(2)λ) =e−∫0t(0)⟨λu,P⟩du−∫t(0)t(1)⟨λu,P(1)(2)⟩du−∫t(1)t(2)⟨λu,P(2)⟩du (2.10) and P(τ0>t0,τ1>t1,τ2>t2) =E[e−∫0t(0)⟨λu,P⟩du−∫t(0)t(1)⟨λu,P(1)(2)⟩du−∫t(1)t(2)⟨λu,P(2)⟩du], (2.11) where $${t_{(j)}}^{'}$$s are ordered times with $$0=t_{(0)}\leq t_{(1)}\leq t_{(2)}$$ and $$(j)$$ refers to the credit name associated with the $$(j+1)$$th ordered time. The following result is a direct consequence of Proposition 2.1. Corollary 2.1 For $$t\geq 0,$$ we have P(τ0∧τ1∧τ2>t|ℑtλ)=e−∫0t⟨λu,P⟩du, and P(τ0∧τ1∧τ2>t)=E[e−∫0t⟨λu,P⟩du]. For $$t_l\geq 0, t_j\geq 0,$$ P(τl>tl,τj>tj)={E[e−∫0tl⟨λu,Plj⟩du−∫tltj⟨λu,Pj⟩du],tl<tj,E[e−∫0tj⟨λu,Plj⟩du−∫tjtl⟨λu,Pl⟩du],tl>tj,E[e−∫0tl⟨λu,Plj⟩du],tl=tj.  The next two results were given in Dong et al. (2014a), which are very helpful for deriving the spread of CDS. Lemma 2.1 Let $$\overline{\tau}=\min\limits_{s\in \mathcal{S}}\tau_s.$$ For any $$\Im-$$measurable random variable $$Y$$ and any $$t>0,$$ we have E[1{τ¯>t}Y|ℑt]=1{τ¯>t}e∫0t⟨λu,P⟩duE[1{τ¯>t}Y|ℑtλ∨ℑtr]. Lemma 2.2 Let $$\overline{\tau}=\min\limits_{s\in \mathcal{S}}\tau_s.$$ Let $$Z$$ be a bounded, $$\Im^r-$$predictable process. Then for any $$0\leq t<\infty,$$ and $$s'\in\mathcal{S},$$ E[Zτs′1{t<τs′≤s}|ℑt]=1{τ¯>t}E[∫tsZuqs′(u)e−∫tuqs′(v)dvdu|ℑtλ∨ℑtr], and E[Zτs′1{τ¯=τs′,t<τs′≤s}|ℑt]=1{τ¯>t}E[∫tsZuqs′(u)e−∫tu⟨λv,P⟩dvdu|ℑtλ∨ℑtr]. 3. Laplace transforms and survival distributions In this section, we shall give the joint Laplace transform of the regime-switching shot noise processes and the integrated regime-switching shot noise processes. Based on the joint Laplace transform, we can obtain the explicit formulas for the joint survival distributions. Define the following process U(t,T)=E[e−∫tTfudu|ℑtX], (3.1) where $$f_{t}=\langle{\bf f}_t,X_{t}\rangle,$$ with $${\bf f}_t=(f_{1}(t),\cdots,f_{N}(t))^{*}.$$ Here, $$f_i(u)$$ is a deterministic function value at $$(0,\infty)$$ satisfying $$\int_0^{\infty}f_i(u)du<\infty,$$ for each $$i = 1,2,\cdots,N.$$ Define $${\bf diag}(\theta)$$ as a diagonal matrix with the diagonal entries given by the vector $$\theta.$$Dong et al. (2014b) gave the expression for $$U(t,T),$$ which is presented in the following Lemma. Lemma 3.1 Let $$U(t,T)$$ be an $$R-$$valued process defined by (3.1). Then we have U(t,T)=∑i=1NUi(t)⟨ei,Xt⟩, (3.2) where $$U_1(t), U_2(t),\ldots, U_N(t)$$ satisfy the following system of $$N$$ coupled ordinary differential equations (ODEs): −fi(t)Ui(t)+dUi(t)dt+∑k=1NqikUk(t)=0,Ui(T)=1,i=1,⋯,N. (3.3) Corollary 3.1 Let $$f_{t}^k=\langle{\bf f}^k_t,X_{t}\rangle$$ for a vector $${\bf f}^k_t=(f_{1}^k(t),\ldots,f_{N}^k(t))^{*}\in R^N,$$ where $$f_i^k(u)$$ is a deterministic function valued at $$(0,\infty)$$ satisfying $$\int_0^{T}f_i^k(u)du<\infty,$$ for each $$k=1,2,i=1,2,\dots,N.$$ For $$\eta\geq 0,$$ then E[e−∫tTfu1du−η∫tTfu2du|ℑtX]=∑i=1NUi0(t,T,η)⟨ei,Xt⟩, (3.4) where $$U^0_i,i=1,\ldots, N$$ are determined by (3.3) with $$f_i(t)$$ replaced by $$f_i^1(t)+\eta f^2_i(t).$$ And E[e−∫tTfu1du∫tTfu2du|ℑtX]=∑i=1NUi1(t,T,0)⟨ei,Xt⟩, (3.5) where $$U^1_i(t,T,0)=-\lim\limits_{\eta\rightarrow 0}\frac{\partial U^0_i(t,T,\eta)}{\partial \eta}.$$ The following result presents the conditional Laplace transform of an integrated regime-switching shot noise process. Lemma 3.2 Let the process $$S_t$$ be defined by (2.2) with $$Y_n\geq 0$$ and $$h>0$$. Then for $$\eta\geq 0,$$ we have E[e−η∫0tSsds|ℑtX]=e−ηS0t+∫0t⟨χs(η),Xs⟩ds, (3.6) where $$\boldsymbol{\chi}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by χsi(η)=λi(∫Rde−ηH(y,t−s)yFi(dy)−1),i=1,…,N. Based on Lemma 3.2, we can derive the conditional joint Laplace transform of a multivariate regime-switching shot noise process and an integrated multivariate regime-switching shot noise process. Corollary 3.2 For $$\eta\geq 0, c^0\geq 0, {\bf c}=(c^1,\ldots,c^m)^*\in R^m, {\bf d}=(d^1,\ldots,d^m)^*\in R^m$$ with $$c^k\geq 0, d^k\geq0$$ for each $$k=1,\ldots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩|ℑtX]=e−(r0+⟨λ0,c⟩)t−η⟨λ0,d⟩+∫0t⟨ϑs(c0,c,d,η),Xs⟩ds, (3.7) where $$\boldsymbol{\vartheta}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by ϑsi(c0,c,d,η)=∑k=1mμki(lski(ck,dk,η)−1)+μ0i(ls0i(c0,0,0)∏k=1mlski(ck,dk,η)−1), (3.8) lski(u,v,η)=∫0∞e−(uHk(t−s)+ηvhk(t−s))yFki(dy),k=0,1,…,m,i=1,…,N and E[⟨λt,d⟩e−∫0t(c0rs+⟨λs,c⟩)ds|ℑtX] =e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds(⟨λ0,d⟩+∫0t⟨F¯s(c0,c,d),Xs⟩ds), (3.9) where $${\bf G}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by Gsi(c0,c)=∑k=1mμki(lski(ck,0,0)−1)+μ0i(∏k=0mlski(ck,0,0)−1). (3.10) And $${\bf \overline{F}}_s$$ is also an $$N$$-dimensional vector with the $$i$$th component given by F¯si(c0,c,d)=∑k=1mμkil¯ski(ck,dk)+μ0i∑k=1m∏l=0,l≠kmlsli(cl,0,0)l¯ski(ck,dk), (3.11) l¯ski(u,v)=vhk(t−s)∫0∞ye−uHk(t−s)yFki(dy),k=1,⋯,m,i=1,⋯,N. The following result is a direct consequence of Corollary 3.2 just by letting $$\eta\rightarrow0$$ in (3.7). Corollary 3.3 For $$c^0\geq 0, {\bf c}=(c^1,\cdots,c^m)^*\in R^m$$ with $$c^k\geq 0,$$ for each $$k=1,\cdots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds|ℑtX]=e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds, (3.12) where $${\bf G}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by (3.10). In particular, when $$F_t^k(y)=\int_0^y\frac{(\beta_k(t))^{n_k}x^{n_k-1}e^{-\beta_k(t)x}}{(n_k-1)!}dx, k=0,1,\cdots,m,$$ where $$\beta_k(t)=\langle\boldsymbol{\beta}_k,X_t\rangle$$ for a constant vector $$\boldsymbol{\beta}_k=(\beta_k^1,\cdots,\beta_k^N)^*\in R^N$$ with $$\beta_k^i>0$$ for each $$i=1,2,\cdots,N,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds|ℑtX]=e−(c0r0+⟨λ0,c⟩)t+∫0t⟨G¯s(c0,c),Xs⟩ds, (3.13) where $${\bf \overline{G}}_s$$ is another $$N$$-dimensional vector with the $$i$$th component given by G¯si(c0,c)=∑k=1mμki((βkickHk(t−s)+βki)nk−1)+μ0i(∏k=0m(βkickHk(t−s)+βki)nk−1). The next result gives the joint Laplace transform of a multivariate regime-switching shot noise process, which plays an important role in deriving the joint survival distributions of the default times and the CDS spread. Proposition 3.1 For $$c^k\geq 0, k=0,1,\ldots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds]=e−(c0r0+⟨λ0,c⟩)t∑i=1NUi2(0,t)⟨ei,X0⟩, (3.14) where $$U^2_i,i=1,\ldots, N$$ satisfy the following system of $$N$$ coupled ODEs: GsiUi2(s)+dUi2(s)ds+∑k=1NqikUk2(s)=0,Ui2(t)=1,i=1,…,N. (3.15) For $$c^0\geq 0, {\bf c}=(c^1,\ldots,c^m)^*\in R^m, {\bf d}=(d^1,\ldots,d^m)^*\in R^m$$ with $$c^i\geq 0, d^i\geq0$$ for each $$i=1,\ldots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds⟨λt,d⟩] =e−(c0r0+⟨λ0,c⟩)t(⟨λ0,d⟩∑i=1NUi2(0,t)⟨ei,X0⟩+∑i=1NUi4(0,t,0)⟨ei,X0⟩), (3.16) where $$U^4_i(0,t,0)=- \lim\limits_{\xi\rightarrow 0}\frac{\partial U^3_i(0,t,\xi)}{\partial \xi},$$ with $$U^3_i(s,t,\xi), i=1,\ldots, N, \xi\geq 0,$$ satisfying the following system of $$N$$ coupled ODEs: (Gsi−ξF¯si)Ui3(s)+dUi3(s)ds+∑k=1NqikUk3(s)=0,Ui3(t,ξ)=1,i=1,…,N. (3.17) From 2.1 and 3.1, we can easily obtain the following formulas for the joint survival distributions of the default times. Corollary 3.4 For $$j\in\{0,1,2\}, t>0,$$ the survival distribution for name $$j$$ is given by P(τj>t)=e−⟨λ0,Pj⟩t∑i=1NΦi1j(0,t)⟨ei,X0⟩, where $$\Phi_i^{1j},i=1,\ldots, N$$ are determined by (3.15) with $$c^0, {\bf c}$$ in $$G_s^i$$ replaced by $$0,{\bf P}^j.$$ For $$l,j\in\{0,1,2\}, t>0,$$ we have P(τl>t,τj>t)=e−⟨λ0,Plj⟩t∑i=1NΦi2lj(0,t)⟨ei,X0⟩, where $$\Phi_i^{2lj},i=1,\ldots, N$$ are determined by (3.15) with $$c^0, {\bf c}$$ in $$G_s^i$$ replaced by $$0,{\bf P}^{lj}.$$ For $$t>0,$$ we have P(τ0>t,τ1>t,τ2>t)=e−⟨λ0,P⟩t∑i=1NΦi3(0,t)⟨ei,X0⟩, where $$\Phi_i^3,i=1,\ldots, N$$ are determined by (3.15) with $$c^0, {\bf c}$$ in $$G_s^i$$ replaced by $$0,{\bf P}.$$ Since the joint distribution and the marginal distributions of the default times have been derived, we can calculate various dependence measures which quantify the relation of pairwise default correlation, such as, the linear correlation coefficient of the default events $$\{\tau_i\leq t\}$$ and $$\{\tau_j\leq t\}:$$ ρtij =P(τi≤t,τj≤t)−P(τi≤t)P(τj≤t)P(τi≤t)(1−P(τi≤t))(P(τj≤t)(1−P(τj≤t)) =P(τi>t,τj>t)−P(τi>t)P(τj>t)P(τi>t)(1−P(τi>t))(P(τj>t)(1−P(τj>t)). Then by using Corollary 3.3, we can obtain the closed-form formula for $$\rho_t^{ij}.$$ 4. CDS with and without counterparty risk In this section, we aim at considering the impact of default risk of the protection seller on the CDS spreads. More precisely, we shall compute the fair CDS premium with and without the default risk of the protection seller and the investor. For simplicity, assume that the face value of the CDS is equal to monetary unit and the spread is paid continuously in time. Let $$T$$ denote the maturity date of the CDS. Let $$\kappa$$ denote the fair spread rate of a CDS contract without the default risk of the protection seller and the protection buyer. Let $$\kappa_1$$ denote the fair spread rate of a CDS contract with counterparty risk. Furthermore, we assume if the protection seller defaults, then the protection buyer gets nothing. There have been many works on discussing the impact of counterparty risk on CDS valuation. In this article, the impact on the CDS spread rate with the presence of the counterparty risk is then measured by $$\kappa_1-\kappa,$$ which is also studied in Leung & Kwok (2009). We first describe the cash flows of a CDS with and without counterparty risk. The cash flows of a CDS without counterparty risk are as follows: Default leg: the protection seller covers the credit losses $$1-R_1$$ as soon as the reference entity has defaulted. Premium leg: the protection buyer pays $$\kappa$$ to the seller continuously, until maturity or until the reference entity defaults before maturity. Then, the fair spread of the CDS without counterparty risk is determined so that the discounted payoff of the two legs are equal when the contract is settled at the initial time. That is, the spread $$\kappa$$ should satisfy κ∫0TE[1{τ1>u}D(0,u)]du=(1−R1)E[D(0,τ1)1{τ1≤T}]. Hence, κ=(1−R1)E[D(0,τ1)1{τ1≤T}]∫0TE[1{τ1>u}D(0,u)]du. (4.1) The cash flows of a CDS with counterparty risk are as follows: Default leg: if the reference entity defaults firstly before maturity, or the reference and the investor default simultaneously before maturity while the protection seller still survives, then the protection seller covers the credit losses $$1-R_1.$$ For simplicity, we assume if the protection seller or the buyer defaults firstly before maturity, then the protection buyer gets nothing. Premium leg: the protection buyer pays $$\kappa_1$$ to the seller continuously, until maturity or until any of names 0, 1 and 2 defaults before maturity. Then, the fair spread of the CDS with counterparty risk is also determined so that the discounted payoff of the two legs are equal when the contract is settled at the initial time. So, the spread $$\kappa_1$$ should satisfy κ1∫0TE[1{τ0∧τ1∧τ2>u}D(0,u)]du=(1−R1)E[D(0,τ1)(1{τ1≤T,τ1<τ2∧τ0}+1{τ1≤T,τ1=τ0<τ2})]. Then, κ1=(1−R1)E[D(0,τ1)(1{τ1≤T,τ1<τ2∧τ0}+1{τ1≤T,τ1=τ0<τ2})]∫0TE[1{τ0∧τ1∧τ2>u}D(0,u)]du. (4.2) Proposition 4.1 The fair CDS premium without counterparty risk is given by κ=(1−R1)∫0Te−(r0+⟨λ0,P1⟩)t∑i=1N(⟨λ0,P1⟩Ψi1(0,t)+Ψi3(0,t,0))⟨ei,X0⟩)dt∫0Te−(r0+⟨λ0,P1⟩)t∑i=1NΨi1(0,t)⟨ei,X0⟩dt, (4.3) where $$\Psi^3_i(0,t,0)=-\lim\limits_{\xi\rightarrow 0}\frac{\partial \Psi^2_i(0,t,\xi)}{\partial \xi},$$$$\Psi^1_i, \Psi^2_i,i=1,\ldots,N$$ are determined by (3.15) and (3.17) with $$c^0,{\bf c},{\bf d}$$ in $$G_s^i, \overline{F}_s^i$$ replaced by $$1,{\bf P}^1$$ and $${\bf P}^1,$$ respectively. Proposition 4.2 Let $${\bf \overline{P}}^{12}=(p_{11}\overline{p}_{12},\ldots,p_{m1}\overline{p}_{m2})^*\in R^m.$$ Then the fair CDS premium with counterparty risk is given by κ1=(1−R1)∫0Te−(r0+⟨λ0,P⟩)t∑i=1N(⟨λ0,P¯12⟩Ψi4(0,t)+Ψi6(0,t,0))⟨ei,X0⟩dt∫0Te−(r0+⟨λ0,P⟩)t∑i=1NΨi4(0,t)⟨ei,X0⟩dt, (4.4) where $$\Psi^6_i(0,t,0)=-\lim\limits_{\xi\rightarrow 0}\frac{\partial \Psi^5_i(0,t,\xi)}{\partial \xi},$$$$\Psi^4_i, \Psi^5_i,i=1,\ldots,N$$ are determined by (3.15) and (3.17) with $$c^0,{\bf c},{\bf d}$$ in $$G_s^i, \overline{F}_s^i$$ replaced by $$1,{\bf P}$$ and $${\bf \overline{P}}^{12}$$, respectively. 5. Numerical results In this section, we shall present some numerical calculations to illustrate our theoretical results. For ease of illustration, we consider $$N = 2,$$ that is $$X$$ switches between only two states, where state $$e_1$$ and state $$e_2$$ represent a ‘good’ economy and a ‘bad’ economy, respectively. The parameters are fixed as follows unless otherwise noted: $$m=4, R_1=R=0.4, T=5, r=(0.05,0.02)^*, p_{1i}=0.1,i=0,1,2,p_{20}=0.1,p_{31}=0.1,p_{42}=0.1,p_{21}=p_{22}=p_{30}=p_{32}=p_{40}=p_{41}=0, \boldsymbol{\lambda_0}^1=(0.01,0.05)^*, \boldsymbol{\lambda_0}^2=(0.015,0.075)^*,\boldsymbol{\lambda_0}^3=(0.025,0.125)^*,\boldsymbol{\lambda_0}^4=(0.02,0.1)^*, \boldsymbol{\mu_0}=(1,5)^*,\boldsymbol{\mu_k}=(1,5)^*,$$ for $$k=1,2,3,4.$$ The density function $$f^k_t$$ is given by $$f^k_t(x)=10e^{-10x}, x>0$$ when $$X_t=e_1,$$ and $$f^k_t(x)=2e^{-2x}, x>0$$ when $$X_t=e_2.$$ Assume $$q_{11}=q_{22}=-q.$$ Furthermore, let $$h^i(t)=e^{-a^it}, i=0,1,\cdots,4,$$ and let $$a^i=50.$$ To investigate the regime-switching effect, we compare the regime-switching intensities model with the one that has no regime switching. So for each $$f_t=\langle{\bf f}, X_t\rangle$$ with $${\bf f}=(f_1,f_2)^*,$$ we choose the constants $$\overline{f}_i$$ in the model without regime switching, such that they satisfy $$e^{-\overline{f}_i T}=E[e^{-\int_0^Tf_{t}dt}|X_{0}=e_i],i=1,2.$$ Figures 1 and 2 plot the impact of the probability $$p_{1i}$$ with $$p_{1i}=p$$ for $$i=0,1,2$$ on the linear correlation coefficient $$\rho_5^{ij}.$$ From Figs 1 and 2, we can see that $$\rho_5^{ij}$$ increases with $$p$$. This is because a larger value of $$p$$ implies an increasing probability of simultaneous defaults. Fig. 1. View largeDownload slide Relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$,$$X_0=e_1,q=0.3$$. Fig. 1. View largeDownload slide Relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$,$$X_0=e_1,q=0.3$$. Fig. 2. View largeDownload slide relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$, $$X_0=e_2,q=0.3$$. Fig. 2. View largeDownload slide relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$, $$X_0=e_2,q=0.3$$. Figures 3 and 4 present the impact of the model parameters on the CDS spread without counterparty risk. From them, we can see the spread corresponding to the case that we start at the ‘good’ economy at time $$t=0$$ is much lower. From Fig. 3, we can conclude that a higher $$q$$ results in a higher spread if $$X_0=e_1.$$ This is because a higher $$q$$ leads to an increasing probability of switching to the ‘bad’ economy. On the other hand, if we start at the ‘bad’ economy, the spreads decrease with $$q$$. This is due to an increasing probability of switching to the ‘good’ economy. We can also see when $$X_0=e_1,$$ the spread in the model without regime switching is higher than that in the regime-switching model, and the reverse relationship holds when $$X_0=e_2.$$ Therefore, if we do not incorporate changes of market regimes into the credit risk modelling, we shall overestimate the spreads during economic expansion and underestimate them during economic recession. From Fig. 4, we can see the impact of the parameter $$a^i$$ on the spread $$\kappa$$ is very obvious, and a higher $$a^i$$ corresponds to a lower spread. That is because when $$a^i$$ increases, the time period that the intensity $$\lambda^i$$ goes back to the previous level of intensity immediately after major events occur will be shorten, and therefore the intensity decreases with $$a^i$$. We can also see the spread increases with $$\boldsymbol{\mu}_0$$ if other parameters are fixed. That is because $$\boldsymbol{\mu}_0$$ increasing indicates the frequency that the intensities jump upwards increases, so that the default intensity of name 1 increases. Fig. 3. View largeDownload slide Relationship between $$\kappa$$ and $$q$$. Fig. 3. View largeDownload slide Relationship between $$\kappa$$ and $$q$$. Fig. 4. View largeDownload slide relationship between $$\kappa$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0, q=0.3$$. Fig. 4. View largeDownload slide relationship between $$\kappa$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0, q=0.3$$. Figures 5 and 6 present the impact of the model parameters on the CDS spread with counterparty risk. The curves of Figs 5 and 6 are similar to those of Figs 3 and 4. From Figs 3–6, we can conclude that the spread with counterparty risk is lower than the one without counterparty risk. Fig. 5. View largeDownload slide Relationship between $$\kappa_1$$ and $$q$$. Fig. 5. View largeDownload slide Relationship between $$\kappa_1$$ and $$q$$. Fig. 6. View largeDownload slide relationship between $$\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. Fig. 6. View largeDownload slide relationship between $$\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. Figures 7 and 8 present the impact of the model parameters on the CDS spread difference $$\kappa-\kappa_1.$$ From Fig. 7 we see, the difference increases with $$q$$ when $$X_0=e_1,$$ while it decreases with $$q$$ when $$X_0=e_2.$$ From Fig. 8, we can conclude the impact of $$a^i$$ and $$\boldsymbol{\mu}_0$$ on $$\kappa-\kappa_1$$ is very obvious. Furthermore, the difference decreases with $$a^i$$ and increases with $$\boldsymbol{\mu}_0$$. Fig. 7. View largeDownload slide Relationship between $$\kappa-\kappa_1$$ and $$q$$. Fig. 7. View largeDownload slide Relationship between $$\kappa-\kappa_1$$ and $$q$$. Fig. 8. View largeDownload slide relationship between $$\kappa-\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. Fig. 8. View largeDownload slide relationship between $$\kappa-\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. To sum up, numerical results indicate that changes of market regimes have a material effect on the spreads. In particular, we can see from Figs 3 and 5, the spread corresponding to the case $$X_0=e_2$$ is much higher than the spread corresponding to the case $$X_0=e_1$$ in the regime-switching model, which is consistent with the economic intuition. This may provide some evidence for justifying the use of the regime-switching model. From Figs 4 and 6, we can conclude that the spread is very sensitive to the jump parameter $$\boldsymbol{\mu}_0$$ and the parameter $$a^i$$, which means incorporating jumps into the intensity processes can provide flexibility for the pricing model. We remark that since this article focuses on providing a theoretical pricing model, we only arbitrarily choose the parameters without doing the calibration in this article. One thing on our future research agenda is to use the credit market CDS spreads and suitable numerical techniques to empirically test our model. Here, we just discuss a feasible method for parameter estimate. The generator of the Markov chain can be borrowed from Giesecke et al. (2011), who suggested that there exist three regimes and obtained the transitional probability by making analysis on the corporate bond market over the course of the last 150 years. For the choice of $$m,$$ we can follow the idea of Lindskog & McNeil (2003). Suppose the three names of the CDS contract can be divided into $$K (K\leq 3)$$ geographical or industry sectors. Given that the names of the CDS contract are subject to idiosyncratic, sector and global shocks, we can assume there are $$m=3+K+1$$ shock event processes. More precisely, for each $$j=1,2,3,$$$$N^j(t)$$ counts the shock events those only trigger the default of name $$j,$$$$N^{4}(t)$$ counts the shock events those trigger the simultaneous defaults of all names, and for each $$l=1,\cdots,K,$$$$N^{4+l}(t)$$ counts the shock events may trigger the defaults of the names belong to their sector. Therefore, for each $$i=1,2,3,j=0,1,2,$$ pij={1,j=i−1,0,j≠i−1  and for each $$j=0,1,2,$$$$p_{4j}=1.$$ If the name $$i$$ belongs to sector $$K_i\in\{1,2,\cdots,K\},$$ then {p4+Kij≠0,j=i,p4+Kij=0,j≠i.  Now the challenging task is to determine the conditional distributions $$F_t^k$$ of the jumps and the functions $$h_k(.),k=0,1,\ldots,m.$$ Since the class of the hyper-exponential distribution is rich enough to approximate many other distributions in the sense of weak convergence, the jumps can be considered to follow a hyper-exponential distribution with regime switching, which has been studied in Corollary 3.1. As is pointed out in 2.1, $$h_k(.)$$ is often set to be of multiplicative type, such that $$h_k(u,t)=ug_k(t).$$ Therefore, it remains to choose suitable functions $$g_k(.), k=0,1,\ldots,m,$$ which are usually set to be decay functions, such as, exponential decay functions and power-decay functions. Once the functions $$g_k(.)$$ are given, the parameter $$\theta=({\bf r_0},\boldsymbol{\lambda_0^i},p_{K_jj},\boldsymbol{\mu_i},{\bf F}^i))$$ for $$i=1,\cdots,m$$ and $$j=0,1,2$$ can be obtained according to θ=argminη^⁡∑T∈{T1,…,Tk}(κ(T,η^)−κ(T))2κ(T)2, where $$T_1,\ldots,T_k$$ are different maturities. We will investigate good methods of parameter estimation to obtain the parameter estimates in the future’s research. 6. Conclusions In this article, we provide an intensity-based model with regime-switching intensities and interest rate to analyse a CDS contract with counterparty credit risk. The model is based on the idea that a firm’s default is driven by idiosyncratic as well as other regional, sectoral, industry or economy-wide shocks, whose arrivals are modelled by a multivariate regime-switching shot noise process. The model captures jumps in interest rate and default intensities as well as the impact of changes of market regimes on their movements over time. Furthermore, it allows us to obtain the joint Laplace transform of the regime-switching shot noise processes. Based on these formulas, we derive the semi-analytic formulas for the CDS spread, which are easy to implement. Numerical results illustrate the regime-switching effects and the jumps have a significant effect on the spread. Therefore, our model might improve the performances of some existing models without jumps or regime switching. One thing on our future research agenda is to empirically test our using market data. Acknowledgements We thank the anonymous referees for the comments which help us to improve this article extensively. Funding QingLan Project (to Y.D.); the National Natural Science Foundation of China (11371274 to G.W.); the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU HKU17329216 to K.C.Y.), and the CAE 2013 research grant from the Society of Actuaries—any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA. Appendix Proof of Proposition 2.1. By using (2.9) and the conditional independence of $$\tau_s, s\in\mathcal{S},$$ we have P(τ0>t0,τ1>t1,τ2>t2|ℑt(2)λ) =P(mins∈S:0∈sτs>t0,mins∈S:1∈sτs>t1,mins∈S:2∈sτs>t0|ℑt(2)λ) =P(⋂j∈{0,1,2}{τ{j}>tj},⋂s={k,l}∈Sτ{k,l}>tk∨tl,τ{0,1,2}>t(2)|ℑt(2)λ) =P(τ{0,1,2}>t(2)|ℑt(2)λ)∏j=02P(τ{j}>tj|ℑt(2)λ)∏s={k,l}∈SP(τs>tk∨tl|ℑt(2)λ) =e−∑j=02∫0tjq{j}(u)du−∑s={k,l}∈S∫0tk∨tlqs(u)ds−∫0t(2)q{0,1,2}(u)du =e−∫0t(0)∑s∈Sqs(u)du−∫t(0)t(1)∑s∈S:s≠{0}qs(u)du−∫t(1)t(2)∑s∈S:(2)∈sqs(u)du. Then substituting the expressions for $$q_{s}(u),s\in\mathcal{S},$$ we can obtain (2.10). Equation (2.11) is an immediate consequence of (2.10). The proof is completed. □ Proof of Corollary 3.1. By making use of (3.2), we can easily obtain equation (3.4). In order to derive equation (3.5), we take the derivative of $$e^{-\int_t^Tf_u^1du-\eta\int_t^Tf_u^2du}$$ with respect to $$\eta$$ and yield ∂e−∫tTfu1du−η∫tTfu2du∂η=−∫tTfu2due−∫tTfu1du−η∫tTfu2du. Note that for each $$k=1,2,i=1,2,\cdots,N,$$$$0<\int_t^Tf_i^k(u)du<\infty$$, and $$e^{-\int_t^Tf_u^1du-\eta\int_t^Tf_u^2du}$$ is bounded by 1. Therefore, the conditional expectation of $$\int_t^Tf_u^2du$$ exists. Then we can interchange the derivative and the expectation, ∂∂ηE[e−∫tTfu1du−η∫tTfu2du|ℑtX]=−E[∫tTfu2due−∫tTfu1du−η∫tTfu2du|ℑtX]. Since $$\int_t^Tf_u^2due^{-\int_t^Tf_u^1du-\eta\int_t^Tf_u^2du}$$ is a continuous and bounded function and the conditional expectation of $$\int_t^Tf_u^2due^{-\int_t^Tf_u^1du}$$ exists, we can interchange the limit and the expectation. That is  limη→0∂∂ηE[e−∫tTfu1du−η∫tTfu2du|ℑtX]=−limη→0E[∫tTfu2due−∫tTfu1du−η∫tTfu2du|ℑtX] =−E[limη→0∫tTfu2due−∫tTfu1du−η∫tTfu2du|ℑtX]=−E[∫tTfu2due−∫tTfu1du|ℑtX]. Therefore, taking derivative with respect to $$\eta$$ on both sides of (3.4) and then letting $$\eta\rightarrow 0$$ gives the result. The proof is finished. □ Proof of Lemma 3.2. By using (2.3), we have E[e−η∫0tSsds|ℑtX]=e−ηS0tE[e−∑Tn≤tηH(Yn,t−Tn)|ℑtX]. Since the path of the Markov chain $$(X_s)_{s\leq t}$$ is known to us, then denote the jump times in the interval $$[0,t]$$ of the Markov chain $$X_s$$ by $$0=\overline{T}_0<\overline{T}_1<\cdots<\overline{T}_k=t.$$ Note that, the processes $$\sum\limits_{\overline{T}_{l-1}< T_j\leq \overline{T}_l}Y_j, l=1,2,\cdots,k$$ are mutually independent compound Poisson processes conditional on $$\Im_t^X$$. Therefore, E[e−∑Tn≤tηH(Yn,t−Tn)|ℑtX]=∏l=1k{E[e−∑T¯l−1<Tn≤T¯lηH(Yn,t−Tn)|Xs=XT¯l−1,T¯l−1≤s<T¯l]}. (A.1) Then from Lemma 3.2 of Gaspar & Schmidt (2010), we have E[e−∑T¯l−1<Tn≤T¯lηH(Yn,t−Tn)|Xs=XT¯l−1,T¯l−1≤s<T¯l] =e∫T¯l−1T¯lλT¯l−1(∫Rde−ηH(y,t−s)yFT¯l−1(dy)−1)ds =e∫T¯l−1T¯lλs(∫Rde−ηH(y,t−s)yFs(dy)−1)ds, (A.2) where the last equality holds because $$\lambda_s=\lambda_{\overline{T}_{l-1}},$$ for $$s\in[\overline{T}_{l-1},\overline{T}_l).$$ Then combining (A.2) with (A.1) concludes the proof. □ Proof of Corollary 3.2. By using (2.6)–(2.8), we have E[e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩|ℑtX]=e−(c0r0+⟨λ0,c⟩)t−η⟨λ0,d⟩ ×E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)−∑k=1m∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX]. Note that, given $$\Im_t^X,$$$$M_0(t), M_1(t),\ldots, M_m(t)$$ are mutually independent, the sequences $$\{Y_1^0,Y_2^0,\ldots\},$$$$\{Y_1^1,Y_2^1,\ldots\},$$$$\ldots,$$$$\{Y_1^m,Y_2^m,\ldots\}$$ are mutually independent, and for each $$k=1,\ldots,m,$$$$Y_1^k,Y_2^k,\ldots$$ are mutually independent. Therefore, E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)−∑k=1m∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX] =E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)|ℑtX] ×∏k=1mE[e−∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX]. From Lemma 3.2, we have E[e−∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX]=e∫0tμk(s)(∫0∞e−(ckHk(t−s)+ηdkhk(t−s))yFsk(dy)−1)ds and E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)|ℑtX] =e∫0tμ0(s)(∫0∞e−c0H0(t−s)yFs0(dy)∏k=1m∫0∞e−(ckHk(t−s)+ηdkhk(t−s))yFsk(dy)−1)ds, which conclude the proof of (3.7). To prove equations (3.9)–(3.11), we should take derivative with respect to $$\eta$$ on both sides of equation (3.7) and then let $$\eta\rightarrow 0.$$ Note that ∂e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩∂η=−⟨λt,d⟩e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩. Since $$r_s\geq 0, \lambda_s\geq 0,$$ the exponential term is continuous and bounded by 1. From the assumptions that $$\int_0^{t}h_k(s)<\infty$$ and $$\int_0^{\infty}yF^{ki}(dy)<\infty,$$ we can conclude the expectation of $$\langle \boldsymbol{\lambda}_t,{\bf d}\rangle$$ exists. Therefore, we can interchange the derivative, the limit and the expectation. The proof is ended. □ Proof of Proposition 3.1. Using the ‘tower property’ of conditional expectations, we have E[e−∫0t(c0rs+⟨λs,c⟩)du] =E[E[e−∫0t(c0rs+⟨λs,c⟩)du|ℑtX]] =E[e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds], where the second equality follows from (3.12). Then using Lemma 3.1 we can easily obtain (3.14). The proof of (3.16) is similar. By the ‘tower property’ of conditional expectations and (3.9), we have E[e−∫0t(c0rs+⟨λs,c⟩)ds⟨λt,d⟩] =E[e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds(⟨λ0,d⟩+∫0t⟨F¯s(c0,c,d),Xs⟩ds)]. Then from Lemma 3.1 and Corollary 3.1, we can give (3.16). The proof is finished. □ Proof of Proposition 4.1. The expected present value of the contingent payment paid by the protection seller from 0 to $$T$$ is given by  (1−R1)E[D(0,τ1)1{τ1≤T}]=(1−R1)E[∫0TD(0,u)1{τ1>u−}dHu1] =(1−R1)∫0TE[e−∫0u(rv+⟨λv,P1⟩)dv⟨λu,P1⟩]du =(1−R1)∫0Te−(r0+⟨λ0,P1⟩)t(⟨λ0,P1⟩∑i=1NΨi1(0,t)⟨ei,X0⟩+∑i=1NΨi3(0,t,0)⟨ei,X0⟩)dt, where the first equality holds because $$H_{t}^1-\int_{0}^{t}1_{\{\tau_{1}>u\}}q_1(u)du$$ is an $$\{\Im_{t}\}$$-martingale, and the last equality follows from (3.16). The total expected present value of the premium payment from 0 to $$T$$ is κ∫0TE[1{τ1>u}D(0,u)]du =κ∫0TE[e−∫0u(rv+⟨λv,P1⟩)dv]du =κ∫0Te−(r0+⟨λ0,P1⟩)t∑i=1NΨi1(0,t)⟨ei,X0⟩dt, where the last equality follows from (3.14). Then substituting the above expressions into (4.1) yields the result. The proof is finished. □ Proof of Proposition 4.2. Let $$\overline{\tau}=\min\limits_{s\in\mathcal{S}}\tau_s.$$ Then the expected present value of the contingent payment paid by the protection seller from 0 to $$T$$ is given by  (1−R1)(E[D(0,τ{1})1{τ¯=τ{1}≤T}]+E[D(0,τ{0,1})1{τ¯=τ{0,1}≤T}]) =(1−R1)∫0TE[e−∫0u(rv+⟨λv,P⟩)dv⟨λu,P¯12⟩]du =(1−R1)∫0Te−(r0+⟨λ0,P⟩)t∑i=1N(⟨λ0,P¯12⟩Ψi4(0,t)+Ψi6(0,t,0))⟨ej,X0⟩dt, where the first equality is obtained by using Lemma 2.2, and the last equality follows from (3.16). The total expected present value of the premium payment from 0 to $$T$$ is κ1∫0TE[1{τ¯>u}D(0,u)]du =κ1∫0TE[e−∫0u(rv+⟨λv,P⟩)dv]du =κ1∫0Te−(r0+⟨λ0,P⟩)t∑i=1NΨi4(0,t)⟨ei,X0⟩dt, where the first equality holds because $$\overline{\tau}$$ has the $$\Im$$-intensity $$\langle \boldsymbol{\lambda}_v,{\bf P}\rangle,$$ and the last equality follows from (3.14). Then substituting the above expressions into (4.2) yields the formula for the spread $$\kappa_1.$$ The proof is completed. □ References Bielecki T. , Crépey S. , Jeanblanc M. & Zargari B. ( 2012 ) Valuation and hedging of CDS counterparty exposure in a Markov copula model. Int. J. Theor. Appl. Finance , 15 , 1 – 39 . Google Scholar Crossref Search ADS Bielecki T. , Cousin A. , Crépey S. & Herbertsson A. ( 2013 ) A bottom-up dynamic model of portfolio credit risk with stochastic intensities and random recoveries, Communication in Statistics-Theory and Methods , 43 , 1362 – 1389 . Google Scholar Crossref Search ADS Black F. & Scholes M. ( 1973 ) The pricing of options and corporate liabilities, Journal of Political Economy , 81 , 637 – 654 . Google Scholar Crossref Search ADS Bo L. J. , Li D. , Wang Y. & Yang X. ( 2013 ) On the conditional default probability in a regulated market with jump risk, Quantitative Finance , 13 , 1967 – 1975 . Google Scholar Crossref Search ADS Cox D. R. & Isham V. ( 1986 ) The virtual waiting time and related processes, Advance in Applied Probabality , 18 , 558 – 573 . Google Scholar Crossref Search ADS Dassios A. & Jang J. ( 2003 ) Pricing of catastrophe reinsurance & derivatives using the Cox process with shot noise intensity, Finance and Stochastics , 7 , 73 – 95 . Google Scholar Crossref Search ADS Davies A. ( 2004 ) Credit spread modeling with regime-switching techniques, Journal of Fixed Income , 14 , 36 – 48 . Google Scholar Crossref Search ADS Davis M. & Lo V. ( 2001 ) Infectious defaults. Quantitative Finance , 1 , 382 – 387 . Google Scholar Crossref Search ADS Di Graziano G. & Rogers L. C. G. ( 2009 ) A dynamic approach to the modelling of correlation credit derivatives using Markov chains. International Journal of Theoretical and Applied Finance , 12 , 45 – 62 . Google Scholar Crossref Search ADS Dong Y. , Yuen K. C. & Wu C. F. ( 2014a ) A multivariate regime-switching mean reverting process and its application to the valuation of credit risk. Stochastic Analysis and Applications , 32 , 687 – 710 . Google Scholar Crossref Search ADS Dong Y. H. , Yuen K. C. & Wu C. F. ( 2014b ) Unilateral counterparty risk valuation of CDS using a regime-switching intensity model. Statistics and Probability Letters , 85 , 25 – 35 . Google Scholar Crossref Search ADS Dong Y. , Yuen K. C. , Wang G. & Wu C. ( 2016 ) A reduced-form model for correlated defaults with regime-switching shot noise intensities. Methodology and Computing in Applied Probability , 18 , 459 – 486 . Google Scholar Crossref Search ADS Duffie D. & Gârleanu N. ( 2001 ) Risk and valuation of collateralized debt obligations. Financial Analysts Journal , 57 , 41 – 59 . Google Scholar Crossref Search ADS Duffie D. & Singleton K. ( 1999 ) Modeling Term Structures of Defaultable Bonds. Review of Financial Studies , 12 , 687 – 720 . Google Scholar Crossref Search ADS Elliott R. J. , Aggoun L. & Moore J. B. ( 1994 ) Hidden Markov Models: Estimation and Control . Berlin-Heidelberg-New York : Springer-Verlag . Gaspar R. M. & Schmidt T. ( 2010 ) Credit risk modeling with shot-noise processes , working paper. Social Science Electronic Publishing . Ge L. , Qiao X. S. & Yue X. Y. ( 2015 ) Explicit formulas for pricing credit-linked notes with counterparty risk under reduced-form framework. IMA Journal of Management Mathematics , 26 , 325 – 344 . Google Scholar Crossref Search ADS Giesecke K. ( 2003 ) A simple exponential model for dependent defaults. Journal of Fixed Income , 13 , 74 – 83 . Google Scholar Crossref Search ADS Giesecke K. , Longstaff F. A. , Schaefer S. & Ilya Strebulaev I. ( 2011 ) Corporate bond default risk: A 150-year perspective. Journal of Financial Economics , 102 , 233 – 250 . Google Scholar Crossref Search ADS Hamilton J. ( 1989 ) A new approach to the economic analysis of nonstationary time series. Econometrica , 57 , 357 – 384 . Google Scholar Crossref Search ADS Hull J. & White A. ( 2004 ) Valuation of a CDO and a nth to default CDS without Monte Carlo simulation. Journal of Derivatives , 12 , 8 – 23 . Google Scholar Crossref Search ADS Jarrow R. & Turnbull T. ( 1995 ) Pricing derivatives on financial securities subject to credit risk. Journal of Finance , 50 , 53 – 85 . Google Scholar Crossref Search ADS Leung K. S. & Kwok Y. K. ( 2009 ) Counterparty risk for credit default swaps: Markov chain interacting intensities model with stochastic intensity. Asia-Pacific Financial Markets , 16 , 169 – 181 . Google Scholar Crossref Search ADS Lindskog F. & McNeil A. ( 2003 ) Common Poisson shock models: applications to insurance and credit risk modelling. Astin Bulletin , 33 , 209 – 238 . Google Scholar Crossref Search ADS Lu Y. , Li S. ( 2009 ) The Markovian regime-switching risk model with a threshold dividend strategy. Insurance: Mathematics and Economics , 44 , 296 – 303 . Merton R. C. ( 1974 ) On the pricing of corporate debt: The risky structure of interest rates, Journal of Finance , 29 , 449 – 470 . Schönbucher P. & Schubert D. ( 2001 ) Copula dependent default risk in intensity models . Working Paper. Department of Statistics, Bonn University . Siu T. K. ( 2010 ) Bond pricing under a Markovian regime-switching jump-augmented Vasicek model via stochastic flows. Applied Mathematics and Computation , 216 , 3184 – 3190 . Google Scholar Crossref Search ADS Zhang X. , Elliott R. J. & Siu T. K. ( 2012 ) A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance. SIAM Journal on Control and Optimization , 50 , 964 – 990 . Google Scholar Crossref Search ADS © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Management Mathematics Oxford University Press

# Correlated default models driven by a multivariate regime-switching shot noise process

, Volume 29 (4) – Sep 7, 2018
25 pages

/lp/ou_press/correlated-default-models-driven-by-a-multivariate-regime-switching-E06vIWiPDs
Publisher
Oxford University Press
ISSN
1471-678X
eISSN
1471-6798
D.O.I.
10.1093/imaman/dpx004
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### Abstract

Abstract We develop a reduced-form credit risk model with regime-switching intensities to investigate the pricing of a credit default swap (CDS) contract. We assume that the defaults of all the names are driven by some shock events. The arrivals of the shock events and the interest rate are modelled by a multivariate regime-switching shot noise process. We provide the flexibility that the model parameters, including the intensities and the jump sizes of the jump component, can switch over time according to a continuous-time, finite-state Markov chain. The states of the chain may be interpreted as different states of an economy or different stages of a business cycle. Based on the joint Laplace transform of the regime-switching shot noise processes, we derive the explicit formulas for the spreads of CDS contract with and without counterparty risk. Numerical results illustrate changes of market regimes have a significant effect on the spread. 1. Introduction In recent years credit derivatives, which are derivatives with a pay-off linked to the credit loss in a portfolio, enjoyed one of the largest growth of all the markets. Credit default swaps (CDS) are the most widely traded form of credit derivatives. A CDS is a financial swap agreement between the buyer of the default protection on a reference risky entity and the seller of the default protection. The protection seller receives fixed periodic payments (CDS spread) from the protection buyer, in return for compensating the buyer’s losses on the reference entity when a credit event occurs. This article focuses on valuing a single-name CDS with and without counterparty risk. There are two primary approaches for pricing credit derivatives, the structural approach and the reduced-form approach. The structural approach, pioneered by Black & Scholes (1973) and Merton (1974), uses the evolution of a firm’s structural variable to determine the time of default, whereas the reduced-form approach, introduced by Jarrow & Turnbull (1995), Duffie & Singleton (1999), treats default as a jump process with an exogenous intensity. Comparing with the structural models, the reduced-form models are more flexible and tractable in dealing with the pricing of multi-name credit derivatives. The works on the reduced-form models differ mainly in specifying default intensities of individual entities and their default dependence. There exist four major approaches to introduce default correlation within the reduced-form framework: the conditionally independent approach, the copula approach, the default contagion models and the common shock models. In the conditionally independent default models, one may set the default intensities of the firms in the portfolio be driven by a common set of macro-economic factors. Therefore, conditional on the realization of the macro-economic state variables, the default times are mutually independent. Some works on the conditionally independent default models include Duffie & Gârleanu (2001), Di Graziano & Rogers (2009), Ge et al. (2015). In the copula models, the dependence structure is linked through a copula function (see, Schönbucher & Schubert, 2001 and Hull & White, 2004). Default contagion is another approach to model the default correlation. The contagion models study the direct interaction of firms in which the default probability of one firm may change upon defaults of some other firms in the portfolio (see Davis & Lo, 2001 and Dong et al., 2016). The common shock models are based on the idea that a firm’s default is driven by exogenous events, for example, policy events, natural catastrophes events, etc. Therefore, simultaneous defaults may occur under the common shock models. Some works on common shock models include Lindskog & McNeil (2003), Giesecke (2003) and Bielecki et al. (2012). This article focuses on the common shock models. In an earlier common shock model, a firm’s default is driven by some idiosyncratic or economy-wide shocks, whose arrivals are modelled by some mutually independent Poisson processes (see, e.g. Lindskog & McNeil, 2003 and Giesecke, 2003). In recent years, there have been numerous contributions on how to model the arrival processes of the shock events by stochastic processes. See, for example, Bielecki et al. (2012) introduced a common shock Markov copula model, in which the arrival intensities are specified in the form of some Cox-Ingersoll-Ross (CIR) model processes. Bielecki et al. (2013) investigated the calibration methodology under a common shock Markov copula model, where the stochastic intensities are specified in the form of an extended CIR process. However, none of the above-mentioned works incorporated jumps into the modelling of the arrival intensities. Shot noise processes, a class of pure jump processes, are particularly useful in the arrival processes since they allow for explicit solutions of many important quantities in derivative pricing. See, for example, Gaspar & Schmidt (2010) consider a multivariate default model driven by shot noise processes and show that a shot noise process gives a superior to historical data as well as a better fit in calibration comparing with an affine jump-diffusion model proposed in Duffie & Gârleanu (2001). The shot noise process (see also, e.g. Cox & Isham, 1986 and Dassios & Jang, 2003) can be well used to measure the impact on the intensities of major events. As was pointed out in Dassios & Jang (2003), the shot noise process measures the frequency, magnitude and time period needed to go back to the previous level of intensity immediately after major events occur. However, empirical studies point to the existence of different regimes in the default risk valuation (see, e.g. Davies, 2004 and Giesecke et al., 2011). Intuitively, the default intensities should be related with the macro-economic conditions since we have witnessed that the recent global financial crisis has had a significant impact on the international financial markets, in particular on the values of credit derivatives. In fact, default risk is much influenced by the business cycles or macro-economy. Default risk typically declines during economic expansion because strong earnings keep overall defaults rates low. Default risk increases during economic recession because earnings deteriorate, making it more difficult to repay loans or make bond payments. Credit derivatives are long term instruments and thus it is very important to develop more appropriate models for valuation and risk management of credit products, which can take into account changes of market regimes due to the crisis. Regime-switching models were introduced by Hamilton (1989) to financial econometrics and economists. One of the main features of these models is that model dynamics are allowed to change over time according to the state of an underlying Markov chain. Regime switches are often interpreted as structural changes in macro-economic conditions and in different stages of business cycles. Recently, regime-switching models have gained immense popularity in the credit risk modelling (see, e.g. Giesecke et al., 2011; Bo et al., 2013; Dong et al., 2014a, 2016). Therefore, motivated by Gaspar & Schmidt (2010), Giesecke et al. (2011), Lindskog & McNeil (2003), we consider a common shock model, in which the arrival intensities of the shock events and the interest rate are governed by a multivariate regime-switching shot noise process. Note that, (Dong et al., 2014a, 2016) also investigate the application of a multivariate regime-switching shot noise process in the credit portfolio model. However, the regime-switching shot noise process considered in Dong et al. (2014a, 2016) is Markovian. More precisely, it is a mean-reverting Markov process or a so-called Ornstein–Uhlenbeck process. By using the Markov property, Dong et al. (2014a, 2016) derive the Laplace transform of the regime-switching shot-noise process via a martingale method. Typically, the regime-switching shot-noise processes are not Markovian. However, in more general cases one looses Markovianity, the martingale method adopted by Dong et al. (2014a, 2016) cannot be applied to derive the Laplace transform of the regime-switching shot noise process. In this article, we consider a more general regime-switching shot-noise process which looses Markovianity and provide an approach to derive the Laplace transform of the regime-switching shot noise process. Summing up, this article aims at providing a flexible and tractable model for correlated defaults which can incorporate the impacts of the changes of market regimes on the interest rate and the default intensities. The contribution of this article is to provide a reduced-form credit risk model with regime-switching shot noise intensities which allows for analytic formulas for the CDS spreads with and without counterparty risk. The structure of the article is organized as follows: in Section 2, we provide the basic setup of our framework, where a common shock model with regime-switching shot noise intensities is introduced and some preliminary results are presented. Section 3 derives the joint Laplace transform of the shot noise processes and the joint survival distributions. In Section 4, we give the closed-form formulas for the CDS spreads with and without counterparty risk. Section 5 performs some numerical experiments to examine how the regime-switching shot noise intensities affect the CDS spread. Section 6 concludes. The proofs are presented in the appendix. 2. Default dependence and some preliminary results In this section, we shall consider a common shocks portfolio credit risk model. Consider a continuous-time model with a finite time horizon $$\mathcal{T}=[0, T]$$ with $$T < \infty$$. Let $$\{\Omega,\Im,\{\Im_t \}_{0\leq t\leq T}, P\}$$ be a filtered complete probability space, where $$P$$ is the risk neutral measure and $$\{\Im_t \}_{0\leq t\leq T}$$ is a filtration satisfying the usual conditions. Throughout the article, it is assumed that all random variables are well defined on this probability space and $$\Im_T-$$measurable. Denote by D(t,T)=e−∫tTrsds the stochastic discount factor at time $$t$$ for maturity $$T,$$ where the stochastic interest rate $$r_t$$ will be specified later. Consider a CDS contract with notional value one, continuous spread rate payments and maturity $$T.$$ Indices 0, 1 and 2 refer to quantities related to the investor, the reference entity and the counterparty. Denote by $$\tau_{0}, \tau_1$$ and $$\tau_{2}$$ be the default times of the investor, the reference entity and the counterparty, respectively; and denote by $$R_{1}$$ the recovery of the reference entity which is supposed to be constant. In this article, we assume that all the cash flows and prices are considered from the perspective of the investor. Now we construct a default dependence structure within the framework of a common shock model. Following Lindskog & McNeil (2003), we assume there are $$m$$ event types called factor names, which can generate events potentially causing joint defaults. Assume that there exists a process $$X:=\{X_t\}_{t\geq 0}$$ which drives all of the arrivals of the $$m$$ types of shock events and the interest rate. Let $$X$$ be a homogenous continuous-time, finite-state, irreducible Markov chain with generator $$Q=(q_{ij})_{i,j=1,2,\cdots,N},$$ generating a filtration $$\Im^X:=\{\Im_t^X|t\in\mathcal{T}\}.$$ The states of the Markov chain $$X$$ are interpreted as different states of a macro-economy or different stages of a business cycle. Without loss of generality, we identify the states of the Markov chain $$X$$ with a set of standard unit vectors $$\mathcal{E}=\{e_1, e_2, \cdots, e_N\},$$ where $$e_i = (0,\cdots, 0, 1, 0,\cdots, 0)^{*}\in R^N,$$$$*$$ denotes the transpose of a vector or a matrix. Elliott et al. (1994) provide the following semi-martingale decomposition for $$X$$: dXt=Q∗Xtdt+dMt, (2.1) where $$\mathcal{M}_{t}$$ is an $$R^N$$-valued martingale with respect to the filtration $$\Im^X$$ under the measure $$P.$$ Assume that the $$m$$ types of shock events arrive as Cox processes $$\{N^{1}(t),t\geq 0\},\cdots,\{N^{m}(t),t\geq 0\}$$ with stochastic intensities $$\lambda^{1}_{t},\cdots, \lambda^{m}_{t}$$ generating a filtration $$\Im^{\boldsymbol{\lambda}}:=\{\Im_t^{\boldsymbol{\lambda}}|t\in\mathcal{T}\},$$ where $$\Im_t^{\boldsymbol{\lambda}}=\Im_t^{\lambda^1}\vee \Im_t^{\lambda^2}\vee\cdots\vee \Im_t^{\lambda^m},$$ with $$\Im_t^{\lambda^k}=\sigma(\lambda_s^k,0\leq s\leq t).$$ Furthermore, assume given $$\Im_t^{\boldsymbol{\lambda}},$$$$\{N^{1}(t),t\geq 0\},\cdots,\{N^{m}(t),t\geq 0\}$$ are mutually independent. Let $$\langle.,.\rangle$$ denote a scalar product in $$R^{N},$$ that is, for any $${\bf x}, {\bf y}\in R^{N},$$$$\langle{\bf x}, {\bf y}\rangle=\sum\limits_{i=1}^{N}x_{i}y_{i}.$$ In order to model the interest rate and the arrival intensities of the shock events, we first introduce the regime-switching shot noise process. Let $$M(t)$$ be a point process with jump times $$(T_n)_{n\geq 1}.$$ If $$X_s=e_i$$ for all $$s$$ in a small interval $$(t,t+h],$$ then $$M(t+h)-M(t)$$ has a Poisson distribution with parameter $$\lambda_i>0.$$ We further assume that given the Markov chain $$X,$$ the process $$\{M(t)\}_{t\geq 0}$$ has independent increments. Then P(M(t+h)=n+1|M(t)=n,Xs=ei,t<s≤t+h)=λih+o(h). The process $$\{M(t)\}_{t\geq 0}$$ is called a regime-switching Poisson process with intensity $$\lambda_t=\langle \boldsymbol{\lambda},X_t\rangle$$ for a constant positive vector $$\boldsymbol{\lambda}=(\overline{\lambda}_1,\cdots,\overline{\lambda}_N)^*\in R^N$$, which is a special case of Cox processes. Let $$Y_1,Y_2,\ldots$$ be a sequence of $$R^d$$-valued random variables. Given $$X_s=e_i,$$ the jump amounts $$Y_1,Y_2,\ldots$$ are independent and identically distributed with distribution function $$F_i(.)$$ and they are independent of $$M(t).$$ The process Jt=∑k=1M(t)Yk is called a regime-switching compound Poisson process, which is widely used in finance and insurance domain (see, e.g. Lu & Li, 2009 and Zhang et al., 2012). Consider a measurable function $$h:R^d\times R\rightarrow R$$ and define the following process by St=S0+∑Tn≤th(Yn,t−Tn)≐S0+Lt, (2.2) where $$S_0=\langle {\bf S}_0, X_0\rangle$$ for a vector $${\bf S}_0=(\overline{S}_1, \overline{S}_2,\cdots, \overline{S}_N)^*\in R^N.$$ Then $$S:=\{S_t\}_{t\geq 0}$$ is a regime-switching shot noise process starting from $$S_0$$ driven by a regime-switching compound Poisson process. If there is no regime switching and $$S_0=0$$, then the model (2.2) is the same as the shot noise process driven by a compound Poisson process, which is investigated in Gaspar & Schmidt (2010). Note that, a regime-switching shot-noise process driven by a regime-switching compound Poisson process can be extended to a regime-switching shot noise process driven by a marked point process with regime switching, which will be considered in our future’s research. Integrating $$S_u$$ from $$0$$ to $$t$$ yields ∫0tSudu=S0t+∫0t∑Tn≤uh(Yn,u−Tn)du=S0t+∑Tn≤tH(Yn,t−Tn), (2.3) where $$H(y,u)=\int_0^uh(y,v)dv.$$ Remark 2.1 As was pointed out in Gaspar & Schmidt (2010), if used for intensity modelling or short interest rate modelling, $$S$$ has to be non-negative. Therefore, we require $$Y_n\geq 0$$ as well as $$h>0.$$ For the choice of $$h,$$ one usually works with $$h$$ of multiplicative type, such that $$h(u,t)=ug(t)$$ for some function $$g$$. In this article, we also consider the case that $$h$$ is of the form of multiplicative type. In particular, (a) if $$g(t)=1,$$ then the process $$L_t$$ in (2.2) becomes a regime-switching compound Poisson process; (b) if $$g(t)=e^{-at},$$ where $$a>0$$ is a constant, then $$L_t$$ is a mean-reverting regime-switching Markov process, and it solves the stochastic differential equation dLt=−aLtdt+dJt. (2.4) As was explained by Dassios & Jang (2003), the process $$L_t$$ in (2.4) measures the frequency, magnitude and time period needed to go back to the previous level of intensity immediately after shock events occur. The process $$L_t$$ decreases until another shock event occurs, which will result in a positive jump in the regime-switching Markov process. Dong et al. (2014a, 2016) considered the pricing of a CDS contract under the model (2.4) via a martingale method. This article will investigate a general regime-switching shot noise process (2.2) by using a different approach. Now, we model the interest rate and the arrival intensities by a multivariate regime-switching shot noise process driven by a multivariate regime-switching compound Poisson process. Assume $$M_0(t),$$$$M_1(t)$$, $$\ldots, M_m(t)$$ are $$m+1$$ conditionally independent regime-switching Poisson processes. For each $$k=0,1,\ldots,m,$$ the intensity of $$M_k(t)$$ is given by $$\mu_k(t)=\langle \boldsymbol{\mu}_k, X_t\rangle,$$ for a positive vector $$\boldsymbol{\mu}_k=(\mu_{k}^{1},\ldots,\mu_{k}^N)^*\in R^N,$$ and denote by $$(T_n^k)_{n\geq 1}$$ the jump times of $$M_k(t).$$ Given $$X_t,$$ let the non-negative jump amounts $$\{Y_1^0, Y_2^0,\ldots\}, \{Y_1^1, Y_2^2,\ldots\},\ldots, \{Y_1^m, Y_2^m,\ldots\}$$ be mutually independent and independent of $$M_k(t),k=0,1,\ldots,m.$$ Assume given $$X_t=e_i,$$$$\{Y_1^k, Y_2^k,\ldots\}$$ is a sequence of independent and identically distributed $$R^+$$-valued random variables with distribution function $$F^{ki}(.)$$ satisfying $$\int_0^{\infty}yF^{ki}(dy)<\infty$$ for each $$k=0,1,\ldots,m,i=1,2,\ldots, N.$$ That is to say, the random variables $$Y_1^k, Y_2^k,\ldots$$ have a conditional distribution $$F_t^k(.)=\langle {\bf F}^k(.),X_t\rangle,$$ where $${\bf F}^k(.)=(F^{k1}(.),\ldots, F^{kN}(.))^*\in R^N.$$ Then the interest rate and the arrival intensities are modelled by rt=r0+∑Tn0≤tYn0h0(t−Tn0), (2.5) and λtk=λ0k+∑Tnk≤tYnkhk(t−Tnk)+∑Tn0≤tYnkhk(t−Tn0),k=1,2…,m. (2.6) Here, $$r_{0}=\langle {\bf r},X_{0}\rangle, \lambda_{0}^k=\langle\boldsymbol{\lambda}^k,X_{0}\rangle,$$ where $${\bf r}=(\overline{r}_{1},\overline{r}_{2},\ldots,\overline{r}_{N})^{*}\in R^N, \boldsymbol{\lambda}^k=(\overline{\lambda}_{1}^k,\overline{\lambda}_{2}^k,\ldots,\overline{\lambda}_{N}^k)^{*}\in R^N$$ with $$\overline{r}_i>0, \overline{\lambda}_i^k>0$$ for each $$k=1,\ldots,m, i=1,2,\ldots,N;$$$$h_l(.)$$ is an $$R^+$$-valued deterministic function satisfying $$\int_0^{t}h_l(s)ds<\infty, \forall t<\infty,$$ for each $$l=0,1,\ldots,m$$. Remark 2.2 Let $$\hat{M}_k(t)=M_k(t)+M_0(t).$$ Then the counting process $$\hat{M}_k(t)$$ is still a regime-switching Poisson process with intensity given by $$\langle \boldsymbol{\mu}_k+\boldsymbol{\mu}_0,X_t\rangle,$$ for each $$k=1,2,\ldots,m.$$ If we denote by $$(\hat{T}_n^k)_{n\geq 1}$$ the jump times of $$\hat{M}_k(t),$$ then (2.6) can be expressed as λtk=λ0k+∑T^nk≤tYnkhk(t−T^nk),k=1,2…,m. Note that, the stochastic interest rate and the intensities $$\lambda^{1}_{t},\ldots, \lambda^{m}_{t}$$ given by (2.5)–(2.6) are driven by a multivariate regime-switching shot noise process with common jumps. As was explained in Siu (2010), the interest rate jumps because of some extraordinary market events, such as market crashes, interventions by central banks or monetary authorities. Therefore, the regime-switching shot noise process can well describe the impact on the interest rate of the major market events. Since extraordinary market events which trigger jumps in the interest rate may also trigger simultaneous jumps in the arrival intensities of the shock events, we assume the intensities are all impacted by $$M_0(t)$$ which counts the number of the arrivals of extraordinary market events. The intensities are still impacted by some factors specified to each group. More precisely, $$M_k(t)$$ only counts the number of the factor events specified to group $$k.$$ Integrating $$r_u$$ from $$0$$ to $$t$$ yields ∫0trudu=r0t+∫0t∑Tn0≤uh0(u−Tn0)Yn0du=r0t+∑Tn0≤tH0(t−Tn0)Yn0, (2.7) where $$H_0(u)=\int_0^uh_0(v)dv.$$ Similarly, ∫0tλukdu=λ0kt+∑Tn0≤tHk(t−Tn0)Ynk+∑Tnk≤tHk(t−Tnk)Ynk,k=1,2,…,m, (2.8) where $$H_k(u)=\int_0^uh_k(v)dv.$$ Now, we construct a default dependence structure by thinning the Cox processes $$N^{e}(t),e=1,\ldots,m.$$ Following Lindskog & McNeil (2003), let $$\{N_j(t), t \geq 0\}$$ be a counting process that counts shocks in the interval $$(0, t]$$ resulting in default of name $$j$$ in $$(0, t], j=0,1,2.$$ At the $$k$$th occurrence of an event of type $$e$$ the Bernoulli variable $$I^{(e)}_{j,k}$$ indicates whether an event of name $$j$$ occurs,$$e=1,2,\ldots,m,j=0,1,2.$$ For simplicity, the vectors Ik(e)=(I0,k(e),I1,k(e),I2,k(e))∗ for $$k = 1, \ldots, N^{e}(t)$$ are considered to be independent and identically distributed with a multivariate Bernoulli distribution. Furthermore, we assume, for each $$k = 1, \ldots, N^{e}(t),$$ the random variables $$I_{0,k}^{(e)},I_{1,k}^{(e)},I_{2,k}^{(e)}$$ are mutually independent. That is, P(Ij1,k(e)=ij1,Ij2,k(e)=ij2)=P(Ij1,k(e)=ij1)P(Ij2,k(e)=ij2),j1,j2∈{0,1,2},ij1,ij2∈{0,1}, and P(I0,k(e)=i0,I1,k(e)=i1,I2,k(e)=i2)=P(I0,k(e)=i0)P(I1,k(e)=i1)P(I2,k(e)=i2),i0,i1,i2∈{0,1}. This article only considers the case that the probability $$P(I_{j,k}^{(e)}=1)$$ is a constant and write for $$P(I_{j,k}^{(e)}=1)=p_{ej},$$ and $$P(I_{j,k}^{(e)}=0)=\overline{p}_{ej},$$ where $$\overline{p}_{ej}=1-p_{ej}.$$ Denote by $$\mathcal{S}$$ the collection of all the non-empty subsets of the set $$\{0, 1, 2\}.$$ For notational convenience, throughout this article, let λt=(λt1,…,λtm)∗∈Rm,Pj=(p1j,…,pmj)∗∈Rm,j=0,1,2,Ps=(∏j∈sp1j∏l∉sp¯1l,…,∏j∈spmj∏l∉sp¯ml)∗∈Rm,s∈S,Plj=(1−p¯1lp¯1j,…,1−p¯mlp¯mj)∗∈Rm.P=(1−∏j=02p¯1j,…,1−∏j=02p¯mj)∗∈Rm. From the above assumptions, the counting process $$N_j(t)$$ can be expressed as Nj(t)=∑e=1m∑k=1Ne(t)Ij,k(e):=∑e=1mNj(e)(t), where $$N^{(e)}_j(t)=\sum\limits_{k=1}^{N^{e}(t)}I_{j,r}^{(e)}.$$ Obviously, given $$\Im_t^{\boldsymbol{\lambda}},$$ the processes $$N^{(e)}_j(t), e=1,\ldots, m$$ are independent non-homogeneous Poisson processes with intensities $$\lambda_t^{e}p_{ej}, e=1,\ldots, m,$$ and the counting process $$N_j(t)$$ is a non-homogeneous Poisson processes with intensity $$q_j(t)=\langle \boldsymbol{\lambda}_t, {\bf P}^j\rangle.$$ By using the process $$N_j(t),$$ we define the default time of name $$j$$ as τj=inf{t≥0:Nj(t)=1},j=0,1,2. So, we have P(τj>t|ℑtλ)=P(Nj(t)=0|ℑtλ)=e−∫0tqj(u)du, and P(τj>t)=P(Nj(t)=0)=E[e−∫0tqj(u)du]. Define the default process as Htj=1{τj≤t},j=0,1,2. Denote the filtration by ℑt=ℑtX∨ℑtr∨ℑtλ∨ℑt0∨ℑt1∨ℑt2, where $$\Im_t^r=\sigma(r_s:0\leq s\leq t),$$ and $$\Im_{t}^{j}=\sigma(H_{u}^{j}:0\leq u\leq t), j=0,1,2.$$ In order to derive the joint survival distribution, we introduce some auxiliary counting processes. For each $$s\in \mathcal{S},$$ define a counting processes $$N_s(t),$$ which counts shocks in the interval $$(0, t]$$ resulting in defaults of all the names in s only. Thus if $$s=\{1,2\},$$ then $$N_s(t)$$ counts shocks which cause simultaneous defaults of names 1 and 2, but not of name 0. Similar to Lindskog & McNeil (2003), we have Ns(t)=∑e=1m∑j=1N(e)(t)∑s′:s′⊇s(−1)|s′|−|s|∏k∈s′Ik,j(e), where $$|s|,s\in \mathcal{S}$$ is the cardinality of the set $$s,$$$$\sum\limits_{s':s'\supseteq s}(-1)^{|s'|-|s|}\prod\limits_{k\in s'}I_{k,j}^{(e)}$$ is an indicator random variable which takes the value 1 if the $$j$$th shock of type $$e$$ causes defaults of all type in $$s$$ only, and the value 0 otherwise. Lindskog & McNeil (2003) proved that when $$N^e(t), e=1,2,\ldots,m$$ are mutually independent Poisson processes, the process $$N_s(t)$$ is a Poisson process and $$\{N_s(t), t \geq 0\}$$ for $$s \in\mathcal{S}$$ are independent Poisson processes. By the same augments as in Lindskog & McNeil (2003), we can conclude that in our model, $$\{N_s(t), t \geq 0\},$$ for $$s \in\mathcal{S},$$ are conditionally independent Cox processes, and for each $$s\in\mathcal{S},$$ the process $$N_s(t)$$ is a Cox process with intensity qs(t)=∑e=1mλte∑s′:s′⊇s(−1)|s′|−|s|∏j∈s′pej. It is easy to check that q{j}(t)=⟨λt,P{j}⟩,{j}∈S,q{l,j}(t)=⟨λt,P{l,j}⟩,{l,j}∈S,q{0,1,2}(t)=⟨λt,P{0,1,2}⟩, and ∑s:s∈Sqs(t)=⟨λt,P⟩. Clearly, we have $$N_j(t)=\sum\limits_{s\in \mathcal{S}:j\in s}N_s(t).$$ In order to derive the joint survival distribution, define the first jump time of $$N_s(t)$$ as τs=inf{t≥0:Ns(t)=1},s∈S. Since $$\{N_s(t), t \geq 0\},$$ for $$s \in\mathcal{S},$$ are conditionally independent Cox processes, the stopping times $$\tau_s,$$ for $$s\in\mathcal{S},$$ are also conditionally independent. Furthermore, the conditional and unconditional survival distributions for $$\tau_s$$ can be expressed as P(τs>t|ℑtλ)=P(Ns(t)=0|ℑtλ)=e−∫0tqs(u)du and P(τs>t)=P(Ns(t)=0)=E[e−∫0tqs(u)du]. Then by using the definition of $$\tau_s,$$ the default time $$\tau_j$$ can be redefined as τj=inf{t≥0:∑s∈S:j∈sNs(t)=1}=mins∈S:j∈sτs,j=0,1,2. (2.9) From (2.9), we can derive the conditional joint survival distribution presented in the following proposition. results. Proposition 2.1 For $$t_j\geq 0, j=0,1,2,$$ we have P(τ0>t0,τ1>t1,τ2>t2|ℑt(2)λ) =e−∫0t(0)⟨λu,P⟩du−∫t(0)t(1)⟨λu,P(1)(2)⟩du−∫t(1)t(2)⟨λu,P(2)⟩du (2.10) and P(τ0>t0,τ1>t1,τ2>t2) =E[e−∫0t(0)⟨λu,P⟩du−∫t(0)t(1)⟨λu,P(1)(2)⟩du−∫t(1)t(2)⟨λu,P(2)⟩du], (2.11) where $${t_{(j)}}^{'}$$s are ordered times with $$0=t_{(0)}\leq t_{(1)}\leq t_{(2)}$$ and $$(j)$$ refers to the credit name associated with the $$(j+1)$$th ordered time. The following result is a direct consequence of Proposition 2.1. Corollary 2.1 For $$t\geq 0,$$ we have P(τ0∧τ1∧τ2>t|ℑtλ)=e−∫0t⟨λu,P⟩du, and P(τ0∧τ1∧τ2>t)=E[e−∫0t⟨λu,P⟩du]. For $$t_l\geq 0, t_j\geq 0,$$ P(τl>tl,τj>tj)={E[e−∫0tl⟨λu,Plj⟩du−∫tltj⟨λu,Pj⟩du],tl<tj,E[e−∫0tj⟨λu,Plj⟩du−∫tjtl⟨λu,Pl⟩du],tl>tj,E[e−∫0tl⟨λu,Plj⟩du],tl=tj.  The next two results were given in Dong et al. (2014a), which are very helpful for deriving the spread of CDS. Lemma 2.1 Let $$\overline{\tau}=\min\limits_{s\in \mathcal{S}}\tau_s.$$ For any $$\Im-$$measurable random variable $$Y$$ and any $$t>0,$$ we have E[1{τ¯>t}Y|ℑt]=1{τ¯>t}e∫0t⟨λu,P⟩duE[1{τ¯>t}Y|ℑtλ∨ℑtr]. Lemma 2.2 Let $$\overline{\tau}=\min\limits_{s\in \mathcal{S}}\tau_s.$$ Let $$Z$$ be a bounded, $$\Im^r-$$predictable process. Then for any $$0\leq t<\infty,$$ and $$s'\in\mathcal{S},$$ E[Zτs′1{t<τs′≤s}|ℑt]=1{τ¯>t}E[∫tsZuqs′(u)e−∫tuqs′(v)dvdu|ℑtλ∨ℑtr], and E[Zτs′1{τ¯=τs′,t<τs′≤s}|ℑt]=1{τ¯>t}E[∫tsZuqs′(u)e−∫tu⟨λv,P⟩dvdu|ℑtλ∨ℑtr]. 3. Laplace transforms and survival distributions In this section, we shall give the joint Laplace transform of the regime-switching shot noise processes and the integrated regime-switching shot noise processes. Based on the joint Laplace transform, we can obtain the explicit formulas for the joint survival distributions. Define the following process U(t,T)=E[e−∫tTfudu|ℑtX], (3.1) where $$f_{t}=\langle{\bf f}_t,X_{t}\rangle,$$ with $${\bf f}_t=(f_{1}(t),\cdots,f_{N}(t))^{*}.$$ Here, $$f_i(u)$$ is a deterministic function value at $$(0,\infty)$$ satisfying $$\int_0^{\infty}f_i(u)du<\infty,$$ for each $$i = 1,2,\cdots,N.$$ Define $${\bf diag}(\theta)$$ as a diagonal matrix with the diagonal entries given by the vector $$\theta.$$Dong et al. (2014b) gave the expression for $$U(t,T),$$ which is presented in the following Lemma. Lemma 3.1 Let $$U(t,T)$$ be an $$R-$$valued process defined by (3.1). Then we have U(t,T)=∑i=1NUi(t)⟨ei,Xt⟩, (3.2) where $$U_1(t), U_2(t),\ldots, U_N(t)$$ satisfy the following system of $$N$$ coupled ordinary differential equations (ODEs): −fi(t)Ui(t)+dUi(t)dt+∑k=1NqikUk(t)=0,Ui(T)=1,i=1,⋯,N. (3.3) Corollary 3.1 Let $$f_{t}^k=\langle{\bf f}^k_t,X_{t}\rangle$$ for a vector $${\bf f}^k_t=(f_{1}^k(t),\ldots,f_{N}^k(t))^{*}\in R^N,$$ where $$f_i^k(u)$$ is a deterministic function valued at $$(0,\infty)$$ satisfying $$\int_0^{T}f_i^k(u)du<\infty,$$ for each $$k=1,2,i=1,2,\dots,N.$$ For $$\eta\geq 0,$$ then E[e−∫tTfu1du−η∫tTfu2du|ℑtX]=∑i=1NUi0(t,T,η)⟨ei,Xt⟩, (3.4) where $$U^0_i,i=1,\ldots, N$$ are determined by (3.3) with $$f_i(t)$$ replaced by $$f_i^1(t)+\eta f^2_i(t).$$ And E[e−∫tTfu1du∫tTfu2du|ℑtX]=∑i=1NUi1(t,T,0)⟨ei,Xt⟩, (3.5) where $$U^1_i(t,T,0)=-\lim\limits_{\eta\rightarrow 0}\frac{\partial U^0_i(t,T,\eta)}{\partial \eta}.$$ The following result presents the conditional Laplace transform of an integrated regime-switching shot noise process. Lemma 3.2 Let the process $$S_t$$ be defined by (2.2) with $$Y_n\geq 0$$ and $$h>0$$. Then for $$\eta\geq 0,$$ we have E[e−η∫0tSsds|ℑtX]=e−ηS0t+∫0t⟨χs(η),Xs⟩ds, (3.6) where $$\boldsymbol{\chi}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by χsi(η)=λi(∫Rde−ηH(y,t−s)yFi(dy)−1),i=1,…,N. Based on Lemma 3.2, we can derive the conditional joint Laplace transform of a multivariate regime-switching shot noise process and an integrated multivariate regime-switching shot noise process. Corollary 3.2 For $$\eta\geq 0, c^0\geq 0, {\bf c}=(c^1,\ldots,c^m)^*\in R^m, {\bf d}=(d^1,\ldots,d^m)^*\in R^m$$ with $$c^k\geq 0, d^k\geq0$$ for each $$k=1,\ldots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩|ℑtX]=e−(r0+⟨λ0,c⟩)t−η⟨λ0,d⟩+∫0t⟨ϑs(c0,c,d,η),Xs⟩ds, (3.7) where $$\boldsymbol{\vartheta}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by ϑsi(c0,c,d,η)=∑k=1mμki(lski(ck,dk,η)−1)+μ0i(ls0i(c0,0,0)∏k=1mlski(ck,dk,η)−1), (3.8) lski(u,v,η)=∫0∞e−(uHk(t−s)+ηvhk(t−s))yFki(dy),k=0,1,…,m,i=1,…,N and E[⟨λt,d⟩e−∫0t(c0rs+⟨λs,c⟩)ds|ℑtX] =e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds(⟨λ0,d⟩+∫0t⟨F¯s(c0,c,d),Xs⟩ds), (3.9) where $${\bf G}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by Gsi(c0,c)=∑k=1mμki(lski(ck,0,0)−1)+μ0i(∏k=0mlski(ck,0,0)−1). (3.10) And $${\bf \overline{F}}_s$$ is also an $$N$$-dimensional vector with the $$i$$th component given by F¯si(c0,c,d)=∑k=1mμkil¯ski(ck,dk)+μ0i∑k=1m∏l=0,l≠kmlsli(cl,0,0)l¯ski(ck,dk), (3.11) l¯ski(u,v)=vhk(t−s)∫0∞ye−uHk(t−s)yFki(dy),k=1,⋯,m,i=1,⋯,N. The following result is a direct consequence of Corollary 3.2 just by letting $$\eta\rightarrow0$$ in (3.7). Corollary 3.3 For $$c^0\geq 0, {\bf c}=(c^1,\cdots,c^m)^*\in R^m$$ with $$c^k\geq 0,$$ for each $$k=1,\cdots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds|ℑtX]=e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds, (3.12) where $${\bf G}_s$$ is an $$N$$-dimensional vector with the $$i$$th component given by (3.10). In particular, when $$F_t^k(y)=\int_0^y\frac{(\beta_k(t))^{n_k}x^{n_k-1}e^{-\beta_k(t)x}}{(n_k-1)!}dx, k=0,1,\cdots,m,$$ where $$\beta_k(t)=\langle\boldsymbol{\beta}_k,X_t\rangle$$ for a constant vector $$\boldsymbol{\beta}_k=(\beta_k^1,\cdots,\beta_k^N)^*\in R^N$$ with $$\beta_k^i>0$$ for each $$i=1,2,\cdots,N,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds|ℑtX]=e−(c0r0+⟨λ0,c⟩)t+∫0t⟨G¯s(c0,c),Xs⟩ds, (3.13) where $${\bf \overline{G}}_s$$ is another $$N$$-dimensional vector with the $$i$$th component given by G¯si(c0,c)=∑k=1mμki((βkickHk(t−s)+βki)nk−1)+μ0i(∏k=0m(βkickHk(t−s)+βki)nk−1). The next result gives the joint Laplace transform of a multivariate regime-switching shot noise process, which plays an important role in deriving the joint survival distributions of the default times and the CDS spread. Proposition 3.1 For $$c^k\geq 0, k=0,1,\ldots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds]=e−(c0r0+⟨λ0,c⟩)t∑i=1NUi2(0,t)⟨ei,X0⟩, (3.14) where $$U^2_i,i=1,\ldots, N$$ satisfy the following system of $$N$$ coupled ODEs: GsiUi2(s)+dUi2(s)ds+∑k=1NqikUk2(s)=0,Ui2(t)=1,i=1,…,N. (3.15) For $$c^0\geq 0, {\bf c}=(c^1,\ldots,c^m)^*\in R^m, {\bf d}=(d^1,\ldots,d^m)^*\in R^m$$ with $$c^i\geq 0, d^i\geq0$$ for each $$i=1,\ldots,m,$$ we have E[e−∫0t(c0rs+⟨λs,c⟩)ds⟨λt,d⟩] =e−(c0r0+⟨λ0,c⟩)t(⟨λ0,d⟩∑i=1NUi2(0,t)⟨ei,X0⟩+∑i=1NUi4(0,t,0)⟨ei,X0⟩), (3.16) where $$U^4_i(0,t,0)=- \lim\limits_{\xi\rightarrow 0}\frac{\partial U^3_i(0,t,\xi)}{\partial \xi},$$ with $$U^3_i(s,t,\xi), i=1,\ldots, N, \xi\geq 0,$$ satisfying the following system of $$N$$ coupled ODEs: (Gsi−ξF¯si)Ui3(s)+dUi3(s)ds+∑k=1NqikUk3(s)=0,Ui3(t,ξ)=1,i=1,…,N. (3.17) From 2.1 and 3.1, we can easily obtain the following formulas for the joint survival distributions of the default times. Corollary 3.4 For $$j\in\{0,1,2\}, t>0,$$ the survival distribution for name $$j$$ is given by P(τj>t)=e−⟨λ0,Pj⟩t∑i=1NΦi1j(0,t)⟨ei,X0⟩, where $$\Phi_i^{1j},i=1,\ldots, N$$ are determined by (3.15) with $$c^0, {\bf c}$$ in $$G_s^i$$ replaced by $$0,{\bf P}^j.$$ For $$l,j\in\{0,1,2\}, t>0,$$ we have P(τl>t,τj>t)=e−⟨λ0,Plj⟩t∑i=1NΦi2lj(0,t)⟨ei,X0⟩, where $$\Phi_i^{2lj},i=1,\ldots, N$$ are determined by (3.15) with $$c^0, {\bf c}$$ in $$G_s^i$$ replaced by $$0,{\bf P}^{lj}.$$ For $$t>0,$$ we have P(τ0>t,τ1>t,τ2>t)=e−⟨λ0,P⟩t∑i=1NΦi3(0,t)⟨ei,X0⟩, where $$\Phi_i^3,i=1,\ldots, N$$ are determined by (3.15) with $$c^0, {\bf c}$$ in $$G_s^i$$ replaced by $$0,{\bf P}.$$ Since the joint distribution and the marginal distributions of the default times have been derived, we can calculate various dependence measures which quantify the relation of pairwise default correlation, such as, the linear correlation coefficient of the default events $$\{\tau_i\leq t\}$$ and $$\{\tau_j\leq t\}:$$ ρtij =P(τi≤t,τj≤t)−P(τi≤t)P(τj≤t)P(τi≤t)(1−P(τi≤t))(P(τj≤t)(1−P(τj≤t)) =P(τi>t,τj>t)−P(τi>t)P(τj>t)P(τi>t)(1−P(τi>t))(P(τj>t)(1−P(τj>t)). Then by using Corollary 3.3, we can obtain the closed-form formula for $$\rho_t^{ij}.$$ 4. CDS with and without counterparty risk In this section, we aim at considering the impact of default risk of the protection seller on the CDS spreads. More precisely, we shall compute the fair CDS premium with and without the default risk of the protection seller and the investor. For simplicity, assume that the face value of the CDS is equal to monetary unit and the spread is paid continuously in time. Let $$T$$ denote the maturity date of the CDS. Let $$\kappa$$ denote the fair spread rate of a CDS contract without the default risk of the protection seller and the protection buyer. Let $$\kappa_1$$ denote the fair spread rate of a CDS contract with counterparty risk. Furthermore, we assume if the protection seller defaults, then the protection buyer gets nothing. There have been many works on discussing the impact of counterparty risk on CDS valuation. In this article, the impact on the CDS spread rate with the presence of the counterparty risk is then measured by $$\kappa_1-\kappa,$$ which is also studied in Leung & Kwok (2009). We first describe the cash flows of a CDS with and without counterparty risk. The cash flows of a CDS without counterparty risk are as follows: Default leg: the protection seller covers the credit losses $$1-R_1$$ as soon as the reference entity has defaulted. Premium leg: the protection buyer pays $$\kappa$$ to the seller continuously, until maturity or until the reference entity defaults before maturity. Then, the fair spread of the CDS without counterparty risk is determined so that the discounted payoff of the two legs are equal when the contract is settled at the initial time. That is, the spread $$\kappa$$ should satisfy κ∫0TE[1{τ1>u}D(0,u)]du=(1−R1)E[D(0,τ1)1{τ1≤T}]. Hence, κ=(1−R1)E[D(0,τ1)1{τ1≤T}]∫0TE[1{τ1>u}D(0,u)]du. (4.1) The cash flows of a CDS with counterparty risk are as follows: Default leg: if the reference entity defaults firstly before maturity, or the reference and the investor default simultaneously before maturity while the protection seller still survives, then the protection seller covers the credit losses $$1-R_1.$$ For simplicity, we assume if the protection seller or the buyer defaults firstly before maturity, then the protection buyer gets nothing. Premium leg: the protection buyer pays $$\kappa_1$$ to the seller continuously, until maturity or until any of names 0, 1 and 2 defaults before maturity. Then, the fair spread of the CDS with counterparty risk is also determined so that the discounted payoff of the two legs are equal when the contract is settled at the initial time. So, the spread $$\kappa_1$$ should satisfy κ1∫0TE[1{τ0∧τ1∧τ2>u}D(0,u)]du=(1−R1)E[D(0,τ1)(1{τ1≤T,τ1<τ2∧τ0}+1{τ1≤T,τ1=τ0<τ2})]. Then, κ1=(1−R1)E[D(0,τ1)(1{τ1≤T,τ1<τ2∧τ0}+1{τ1≤T,τ1=τ0<τ2})]∫0TE[1{τ0∧τ1∧τ2>u}D(0,u)]du. (4.2) Proposition 4.1 The fair CDS premium without counterparty risk is given by κ=(1−R1)∫0Te−(r0+⟨λ0,P1⟩)t∑i=1N(⟨λ0,P1⟩Ψi1(0,t)+Ψi3(0,t,0))⟨ei,X0⟩)dt∫0Te−(r0+⟨λ0,P1⟩)t∑i=1NΨi1(0,t)⟨ei,X0⟩dt, (4.3) where $$\Psi^3_i(0,t,0)=-\lim\limits_{\xi\rightarrow 0}\frac{\partial \Psi^2_i(0,t,\xi)}{\partial \xi},$$$$\Psi^1_i, \Psi^2_i,i=1,\ldots,N$$ are determined by (3.15) and (3.17) with $$c^0,{\bf c},{\bf d}$$ in $$G_s^i, \overline{F}_s^i$$ replaced by $$1,{\bf P}^1$$ and $${\bf P}^1,$$ respectively. Proposition 4.2 Let $${\bf \overline{P}}^{12}=(p_{11}\overline{p}_{12},\ldots,p_{m1}\overline{p}_{m2})^*\in R^m.$$ Then the fair CDS premium with counterparty risk is given by κ1=(1−R1)∫0Te−(r0+⟨λ0,P⟩)t∑i=1N(⟨λ0,P¯12⟩Ψi4(0,t)+Ψi6(0,t,0))⟨ei,X0⟩dt∫0Te−(r0+⟨λ0,P⟩)t∑i=1NΨi4(0,t)⟨ei,X0⟩dt, (4.4) where $$\Psi^6_i(0,t,0)=-\lim\limits_{\xi\rightarrow 0}\frac{\partial \Psi^5_i(0,t,\xi)}{\partial \xi},$$$$\Psi^4_i, \Psi^5_i,i=1,\ldots,N$$ are determined by (3.15) and (3.17) with $$c^0,{\bf c},{\bf d}$$ in $$G_s^i, \overline{F}_s^i$$ replaced by $$1,{\bf P}$$ and $${\bf \overline{P}}^{12}$$, respectively. 5. Numerical results In this section, we shall present some numerical calculations to illustrate our theoretical results. For ease of illustration, we consider $$N = 2,$$ that is $$X$$ switches between only two states, where state $$e_1$$ and state $$e_2$$ represent a ‘good’ economy and a ‘bad’ economy, respectively. The parameters are fixed as follows unless otherwise noted: $$m=4, R_1=R=0.4, T=5, r=(0.05,0.02)^*, p_{1i}=0.1,i=0,1,2,p_{20}=0.1,p_{31}=0.1,p_{42}=0.1,p_{21}=p_{22}=p_{30}=p_{32}=p_{40}=p_{41}=0, \boldsymbol{\lambda_0}^1=(0.01,0.05)^*, \boldsymbol{\lambda_0}^2=(0.015,0.075)^*,\boldsymbol{\lambda_0}^3=(0.025,0.125)^*,\boldsymbol{\lambda_0}^4=(0.02,0.1)^*, \boldsymbol{\mu_0}=(1,5)^*,\boldsymbol{\mu_k}=(1,5)^*,$$ for $$k=1,2,3,4.$$ The density function $$f^k_t$$ is given by $$f^k_t(x)=10e^{-10x}, x>0$$ when $$X_t=e_1,$$ and $$f^k_t(x)=2e^{-2x}, x>0$$ when $$X_t=e_2.$$ Assume $$q_{11}=q_{22}=-q.$$ Furthermore, let $$h^i(t)=e^{-a^it}, i=0,1,\cdots,4,$$ and let $$a^i=50.$$ To investigate the regime-switching effect, we compare the regime-switching intensities model with the one that has no regime switching. So for each $$f_t=\langle{\bf f}, X_t\rangle$$ with $${\bf f}=(f_1,f_2)^*,$$ we choose the constants $$\overline{f}_i$$ in the model without regime switching, such that they satisfy $$e^{-\overline{f}_i T}=E[e^{-\int_0^Tf_{t}dt}|X_{0}=e_i],i=1,2.$$ Figures 1 and 2 plot the impact of the probability $$p_{1i}$$ with $$p_{1i}=p$$ for $$i=0,1,2$$ on the linear correlation coefficient $$\rho_5^{ij}.$$ From Figs 1 and 2, we can see that $$\rho_5^{ij}$$ increases with $$p$$. This is because a larger value of $$p$$ implies an increasing probability of simultaneous defaults. Fig. 1. View largeDownload slide Relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$,$$X_0=e_1,q=0.3$$. Fig. 1. View largeDownload slide Relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$,$$X_0=e_1,q=0.3$$. Fig. 2. View largeDownload slide relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$, $$X_0=e_2,q=0.3$$. Fig. 2. View largeDownload slide relationship between $$\rho_5^{ij}$$ and $$p(p_{1i}=p,i=0,1,2)$$, $$X_0=e_2,q=0.3$$. Figures 3 and 4 present the impact of the model parameters on the CDS spread without counterparty risk. From them, we can see the spread corresponding to the case that we start at the ‘good’ economy at time $$t=0$$ is much lower. From Fig. 3, we can conclude that a higher $$q$$ results in a higher spread if $$X_0=e_1.$$ This is because a higher $$q$$ leads to an increasing probability of switching to the ‘bad’ economy. On the other hand, if we start at the ‘bad’ economy, the spreads decrease with $$q$$. This is due to an increasing probability of switching to the ‘good’ economy. We can also see when $$X_0=e_1,$$ the spread in the model without regime switching is higher than that in the regime-switching model, and the reverse relationship holds when $$X_0=e_2.$$ Therefore, if we do not incorporate changes of market regimes into the credit risk modelling, we shall overestimate the spreads during economic expansion and underestimate them during economic recession. From Fig. 4, we can see the impact of the parameter $$a^i$$ on the spread $$\kappa$$ is very obvious, and a higher $$a^i$$ corresponds to a lower spread. That is because when $$a^i$$ increases, the time period that the intensity $$\lambda^i$$ goes back to the previous level of intensity immediately after major events occur will be shorten, and therefore the intensity decreases with $$a^i$$. We can also see the spread increases with $$\boldsymbol{\mu}_0$$ if other parameters are fixed. That is because $$\boldsymbol{\mu}_0$$ increasing indicates the frequency that the intensities jump upwards increases, so that the default intensity of name 1 increases. Fig. 3. View largeDownload slide Relationship between $$\kappa$$ and $$q$$. Fig. 3. View largeDownload slide Relationship between $$\kappa$$ and $$q$$. Fig. 4. View largeDownload slide relationship between $$\kappa$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0, q=0.3$$. Fig. 4. View largeDownload slide relationship between $$\kappa$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0, q=0.3$$. Figures 5 and 6 present the impact of the model parameters on the CDS spread with counterparty risk. The curves of Figs 5 and 6 are similar to those of Figs 3 and 4. From Figs 3–6, we can conclude that the spread with counterparty risk is lower than the one without counterparty risk. Fig. 5. View largeDownload slide Relationship between $$\kappa_1$$ and $$q$$. Fig. 5. View largeDownload slide Relationship between $$\kappa_1$$ and $$q$$. Fig. 6. View largeDownload slide relationship between $$\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. Fig. 6. View largeDownload slide relationship between $$\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. Figures 7 and 8 present the impact of the model parameters on the CDS spread difference $$\kappa-\kappa_1.$$ From Fig. 7 we see, the difference increases with $$q$$ when $$X_0=e_1,$$ while it decreases with $$q$$ when $$X_0=e_2.$$ From Fig. 8, we can conclude the impact of $$a^i$$ and $$\boldsymbol{\mu}_0$$ on $$\kappa-\kappa_1$$ is very obvious. Furthermore, the difference decreases with $$a^i$$ and increases with $$\boldsymbol{\mu}_0$$. Fig. 7. View largeDownload slide Relationship between $$\kappa-\kappa_1$$ and $$q$$. Fig. 7. View largeDownload slide Relationship between $$\kappa-\kappa_1$$ and $$q$$. Fig. 8. View largeDownload slide relationship between $$\kappa-\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. Fig. 8. View largeDownload slide relationship between $$\kappa-\kappa_1$$ and $$a^i$$ for different $$\boldsymbol{\mu}_0$$ and $$X_0,q=0.3$$. To sum up, numerical results indicate that changes of market regimes have a material effect on the spreads. In particular, we can see from Figs 3 and 5, the spread corresponding to the case $$X_0=e_2$$ is much higher than the spread corresponding to the case $$X_0=e_1$$ in the regime-switching model, which is consistent with the economic intuition. This may provide some evidence for justifying the use of the regime-switching model. From Figs 4 and 6, we can conclude that the spread is very sensitive to the jump parameter $$\boldsymbol{\mu}_0$$ and the parameter $$a^i$$, which means incorporating jumps into the intensity processes can provide flexibility for the pricing model. We remark that since this article focuses on providing a theoretical pricing model, we only arbitrarily choose the parameters without doing the calibration in this article. One thing on our future research agenda is to use the credit market CDS spreads and suitable numerical techniques to empirically test our model. Here, we just discuss a feasible method for parameter estimate. The generator of the Markov chain can be borrowed from Giesecke et al. (2011), who suggested that there exist three regimes and obtained the transitional probability by making analysis on the corporate bond market over the course of the last 150 years. For the choice of $$m,$$ we can follow the idea of Lindskog & McNeil (2003). Suppose the three names of the CDS contract can be divided into $$K (K\leq 3)$$ geographical or industry sectors. Given that the names of the CDS contract are subject to idiosyncratic, sector and global shocks, we can assume there are $$m=3+K+1$$ shock event processes. More precisely, for each $$j=1,2,3,$$$$N^j(t)$$ counts the shock events those only trigger the default of name $$j,$$$$N^{4}(t)$$ counts the shock events those trigger the simultaneous defaults of all names, and for each $$l=1,\cdots,K,$$$$N^{4+l}(t)$$ counts the shock events may trigger the defaults of the names belong to their sector. Therefore, for each $$i=1,2,3,j=0,1,2,$$ pij={1,j=i−1,0,j≠i−1  and for each $$j=0,1,2,$$$$p_{4j}=1.$$ If the name $$i$$ belongs to sector $$K_i\in\{1,2,\cdots,K\},$$ then {p4+Kij≠0,j=i,p4+Kij=0,j≠i.  Now the challenging task is to determine the conditional distributions $$F_t^k$$ of the jumps and the functions $$h_k(.),k=0,1,\ldots,m.$$ Since the class of the hyper-exponential distribution is rich enough to approximate many other distributions in the sense of weak convergence, the jumps can be considered to follow a hyper-exponential distribution with regime switching, which has been studied in Corollary 3.1. As is pointed out in 2.1, $$h_k(.)$$ is often set to be of multiplicative type, such that $$h_k(u,t)=ug_k(t).$$ Therefore, it remains to choose suitable functions $$g_k(.), k=0,1,\ldots,m,$$ which are usually set to be decay functions, such as, exponential decay functions and power-decay functions. Once the functions $$g_k(.)$$ are given, the parameter $$\theta=({\bf r_0},\boldsymbol{\lambda_0^i},p_{K_jj},\boldsymbol{\mu_i},{\bf F}^i))$$ for $$i=1,\cdots,m$$ and $$j=0,1,2$$ can be obtained according to θ=argminη^⁡∑T∈{T1,…,Tk}(κ(T,η^)−κ(T))2κ(T)2, where $$T_1,\ldots,T_k$$ are different maturities. We will investigate good methods of parameter estimation to obtain the parameter estimates in the future’s research. 6. Conclusions In this article, we provide an intensity-based model with regime-switching intensities and interest rate to analyse a CDS contract with counterparty credit risk. The model is based on the idea that a firm’s default is driven by idiosyncratic as well as other regional, sectoral, industry or economy-wide shocks, whose arrivals are modelled by a multivariate regime-switching shot noise process. The model captures jumps in interest rate and default intensities as well as the impact of changes of market regimes on their movements over time. Furthermore, it allows us to obtain the joint Laplace transform of the regime-switching shot noise processes. Based on these formulas, we derive the semi-analytic formulas for the CDS spread, which are easy to implement. Numerical results illustrate the regime-switching effects and the jumps have a significant effect on the spread. Therefore, our model might improve the performances of some existing models without jumps or regime switching. One thing on our future research agenda is to empirically test our using market data. Acknowledgements We thank the anonymous referees for the comments which help us to improve this article extensively. Funding QingLan Project (to Y.D.); the National Natural Science Foundation of China (11371274 to G.W.); the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU HKU17329216 to K.C.Y.), and the CAE 2013 research grant from the Society of Actuaries—any opinions, finding, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the SOA. Appendix Proof of Proposition 2.1. By using (2.9) and the conditional independence of $$\tau_s, s\in\mathcal{S},$$ we have P(τ0>t0,τ1>t1,τ2>t2|ℑt(2)λ) =P(mins∈S:0∈sτs>t0,mins∈S:1∈sτs>t1,mins∈S:2∈sτs>t0|ℑt(2)λ) =P(⋂j∈{0,1,2}{τ{j}>tj},⋂s={k,l}∈Sτ{k,l}>tk∨tl,τ{0,1,2}>t(2)|ℑt(2)λ) =P(τ{0,1,2}>t(2)|ℑt(2)λ)∏j=02P(τ{j}>tj|ℑt(2)λ)∏s={k,l}∈SP(τs>tk∨tl|ℑt(2)λ) =e−∑j=02∫0tjq{j}(u)du−∑s={k,l}∈S∫0tk∨tlqs(u)ds−∫0t(2)q{0,1,2}(u)du =e−∫0t(0)∑s∈Sqs(u)du−∫t(0)t(1)∑s∈S:s≠{0}qs(u)du−∫t(1)t(2)∑s∈S:(2)∈sqs(u)du. Then substituting the expressions for $$q_{s}(u),s\in\mathcal{S},$$ we can obtain (2.10). Equation (2.11) is an immediate consequence of (2.10). The proof is completed. □ Proof of Corollary 3.1. By making use of (3.2), we can easily obtain equation (3.4). In order to derive equation (3.5), we take the derivative of $$e^{-\int_t^Tf_u^1du-\eta\int_t^Tf_u^2du}$$ with respect to $$\eta$$ and yield ∂e−∫tTfu1du−η∫tTfu2du∂η=−∫tTfu2due−∫tTfu1du−η∫tTfu2du. Note that for each $$k=1,2,i=1,2,\cdots,N,$$$$0<\int_t^Tf_i^k(u)du<\infty$$, and $$e^{-\int_t^Tf_u^1du-\eta\int_t^Tf_u^2du}$$ is bounded by 1. Therefore, the conditional expectation of $$\int_t^Tf_u^2du$$ exists. Then we can interchange the derivative and the expectation, ∂∂ηE[e−∫tTfu1du−η∫tTfu2du|ℑtX]=−E[∫tTfu2due−∫tTfu1du−η∫tTfu2du|ℑtX]. Since $$\int_t^Tf_u^2due^{-\int_t^Tf_u^1du-\eta\int_t^Tf_u^2du}$$ is a continuous and bounded function and the conditional expectation of $$\int_t^Tf_u^2due^{-\int_t^Tf_u^1du}$$ exists, we can interchange the limit and the expectation. That is  limη→0∂∂ηE[e−∫tTfu1du−η∫tTfu2du|ℑtX]=−limη→0E[∫tTfu2due−∫tTfu1du−η∫tTfu2du|ℑtX] =−E[limη→0∫tTfu2due−∫tTfu1du−η∫tTfu2du|ℑtX]=−E[∫tTfu2due−∫tTfu1du|ℑtX]. Therefore, taking derivative with respect to $$\eta$$ on both sides of (3.4) and then letting $$\eta\rightarrow 0$$ gives the result. The proof is finished. □ Proof of Lemma 3.2. By using (2.3), we have E[e−η∫0tSsds|ℑtX]=e−ηS0tE[e−∑Tn≤tηH(Yn,t−Tn)|ℑtX]. Since the path of the Markov chain $$(X_s)_{s\leq t}$$ is known to us, then denote the jump times in the interval $$[0,t]$$ of the Markov chain $$X_s$$ by $$0=\overline{T}_0<\overline{T}_1<\cdots<\overline{T}_k=t.$$ Note that, the processes $$\sum\limits_{\overline{T}_{l-1}< T_j\leq \overline{T}_l}Y_j, l=1,2,\cdots,k$$ are mutually independent compound Poisson processes conditional on $$\Im_t^X$$. Therefore, E[e−∑Tn≤tηH(Yn,t−Tn)|ℑtX]=∏l=1k{E[e−∑T¯l−1<Tn≤T¯lηH(Yn,t−Tn)|Xs=XT¯l−1,T¯l−1≤s<T¯l]}. (A.1) Then from Lemma 3.2 of Gaspar & Schmidt (2010), we have E[e−∑T¯l−1<Tn≤T¯lηH(Yn,t−Tn)|Xs=XT¯l−1,T¯l−1≤s<T¯l] =e∫T¯l−1T¯lλT¯l−1(∫Rde−ηH(y,t−s)yFT¯l−1(dy)−1)ds =e∫T¯l−1T¯lλs(∫Rde−ηH(y,t−s)yFs(dy)−1)ds, (A.2) where the last equality holds because $$\lambda_s=\lambda_{\overline{T}_{l-1}},$$ for $$s\in[\overline{T}_{l-1},\overline{T}_l).$$ Then combining (A.2) with (A.1) concludes the proof. □ Proof of Corollary 3.2. By using (2.6)–(2.8), we have E[e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩|ℑtX]=e−(c0r0+⟨λ0,c⟩)t−η⟨λ0,d⟩ ×E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)−∑k=1m∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX]. Note that, given $$\Im_t^X,$$$$M_0(t), M_1(t),\ldots, M_m(t)$$ are mutually independent, the sequences $$\{Y_1^0,Y_2^0,\ldots\},$$$$\{Y_1^1,Y_2^1,\ldots\},$$$$\ldots,$$$$\{Y_1^m,Y_2^m,\ldots\}$$ are mutually independent, and for each $$k=1,\ldots,m,$$$$Y_1^k,Y_2^k,\ldots$$ are mutually independent. Therefore, E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)−∑k=1m∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX] =E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)|ℑtX] ×∏k=1mE[e−∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX]. From Lemma 3.2, we have E[e−∑Tnk≤t(ckHk(t−Tnk)+ηdkhk(t−Tnk))Ynk|ℑtX]=e∫0tμk(s)(∫0∞e−(ckHk(t−s)+ηdkhk(t−s))yFsk(dy)−1)ds and E[e−∑Tn0≤t(c0H0(t−Tn0)Yn0+∑k=1m(ckHk(t−Tn0)+ηdkhk(t−Tn0))Ynk)|ℑtX] =e∫0tμ0(s)(∫0∞e−c0H0(t−s)yFs0(dy)∏k=1m∫0∞e−(ckHk(t−s)+ηdkhk(t−s))yFsk(dy)−1)ds, which conclude the proof of (3.7). To prove equations (3.9)–(3.11), we should take derivative with respect to $$\eta$$ on both sides of equation (3.7) and then let $$\eta\rightarrow 0.$$ Note that ∂e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩∂η=−⟨λt,d⟩e−∫0t(c0rs+⟨λs,c⟩)ds−η⟨λt,d⟩. Since $$r_s\geq 0, \lambda_s\geq 0,$$ the exponential term is continuous and bounded by 1. From the assumptions that $$\int_0^{t}h_k(s)<\infty$$ and $$\int_0^{\infty}yF^{ki}(dy)<\infty,$$ we can conclude the expectation of $$\langle \boldsymbol{\lambda}_t,{\bf d}\rangle$$ exists. Therefore, we can interchange the derivative, the limit and the expectation. The proof is ended. □ Proof of Proposition 3.1. Using the ‘tower property’ of conditional expectations, we have E[e−∫0t(c0rs+⟨λs,c⟩)du] =E[E[e−∫0t(c0rs+⟨λs,c⟩)du|ℑtX]] =E[e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds], where the second equality follows from (3.12). Then using Lemma 3.1 we can easily obtain (3.14). The proof of (3.16) is similar. By the ‘tower property’ of conditional expectations and (3.9), we have E[e−∫0t(c0rs+⟨λs,c⟩)ds⟨λt,d⟩] =E[e−(c0r0+⟨λ0,c⟩)t+∫0t⟨Gs(c0,c),Xs⟩ds(⟨λ0,d⟩+∫0t⟨F¯s(c0,c,d),Xs⟩ds)]. Then from Lemma 3.1 and Corollary 3.1, we can give (3.16). The proof is finished. □ Proof of Proposition 4.1. The expected present value of the contingent payment paid by the protection seller from 0 to $$T$$ is given by  (1−R1)E[D(0,τ1)1{τ1≤T}]=(1−R1)E[∫0TD(0,u)1{τ1>u−}dHu1] =(1−R1)∫0TE[e−∫0u(rv+⟨λv,P1⟩)dv⟨λu,P1⟩]du =(1−R1)∫0Te−(r0+⟨λ0,P1⟩)t(⟨λ0,P1⟩∑i=1NΨi1(0,t)⟨ei,X0⟩+∑i=1NΨi3(0,t,0)⟨ei,X0⟩)dt, where the first equality holds because $$H_{t}^1-\int_{0}^{t}1_{\{\tau_{1}>u\}}q_1(u)du$$ is an $$\{\Im_{t}\}$$-martingale, and the last equality follows from (3.16). The total expected present value of the premium payment from 0 to $$T$$ is κ∫0TE[1{τ1>u}D(0,u)]du =κ∫0TE[e−∫0u(rv+⟨λv,P1⟩)dv]du =κ∫0Te−(r0+⟨λ0,P1⟩)t∑i=1NΨi1(0,t)⟨ei,X0⟩dt, where the last equality follows from (3.14). Then substituting the above expressions into (4.1) yields the result. The proof is finished. □ Proof of Proposition 4.2. Let $$\overline{\tau}=\min\limits_{s\in\mathcal{S}}\tau_s.$$ Then the expected present value of the contingent payment paid by the protection seller from 0 to $$T$$ is given by  (1−R1)(E[D(0,τ{1})1{τ¯=τ{1}≤T}]+E[D(0,τ{0,1})1{τ¯=τ{0,1}≤T}]) =(1−R1)∫0TE[e−∫0u(rv+⟨λv,P⟩)dv⟨λu,P¯12⟩]du =(1−R1)∫0Te−(r0+⟨λ0,P⟩)t∑i=1N(⟨λ0,P¯12⟩Ψi4(0,t)+Ψi6(0,t,0))⟨ej,X0⟩dt, where the first equality is obtained by using Lemma 2.2, and the last equality follows from (3.16). The total expected present value of the premium payment from 0 to $$T$$ is κ1∫0TE[1{τ¯>u}D(0,u)]du =κ1∫0TE[e−∫0u(rv+⟨λv,P⟩)dv]du =κ1∫0Te−(r0+⟨λ0,P⟩)t∑i=1NΨi4(0,t)⟨ei,X0⟩dt, where the first equality holds because $$\overline{\tau}$$ has the $$\Im$$-intensity $$\langle \boldsymbol{\lambda}_v,{\bf P}\rangle,$$ and the last equality follows from (3.14). Then substituting the above expressions into (4.2) yields the formula for the spread $$\kappa_1.$$ The proof is completed. □ References Bielecki T. , Crépey S. , Jeanblanc M. & Zargari B. ( 2012 ) Valuation and hedging of CDS counterparty exposure in a Markov copula model. Int. J. Theor. Appl. Finance , 15 , 1 – 39 . Google Scholar Crossref Search ADS Bielecki T. , Cousin A. , Crépey S. & Herbertsson A. ( 2013 ) A bottom-up dynamic model of portfolio credit risk with stochastic intensities and random recoveries, Communication in Statistics-Theory and Methods , 43 , 1362 – 1389 . Google Scholar Crossref Search ADS Black F. & Scholes M. ( 1973 ) The pricing of options and corporate liabilities, Journal of Political Economy , 81 , 637 – 654 . Google Scholar Crossref Search ADS Bo L. J. , Li D. , Wang Y. & Yang X. ( 2013 ) On the conditional default probability in a regulated market with jump risk, Quantitative Finance , 13 , 1967 – 1975 . Google Scholar Crossref Search ADS Cox D. R. & Isham V. ( 1986 ) The virtual waiting time and related processes, Advance in Applied Probabality , 18 , 558 – 573 . Google Scholar Crossref Search ADS Dassios A. & Jang J. ( 2003 ) Pricing of catastrophe reinsurance & derivatives using the Cox process with shot noise intensity, Finance and Stochastics , 7 , 73 – 95 . Google Scholar Crossref Search ADS Davies A. ( 2004 ) Credit spread modeling with regime-switching techniques, Journal of Fixed Income , 14 , 36 – 48 . Google Scholar Crossref Search ADS Davis M. & Lo V. ( 2001 ) Infectious defaults. Quantitative Finance , 1 , 382 – 387 . Google Scholar Crossref Search ADS Di Graziano G. & Rogers L. C. G. ( 2009 ) A dynamic approach to the modelling of correlation credit derivatives using Markov chains. International Journal of Theoretical and Applied Finance , 12 , 45 – 62 . Google Scholar Crossref Search ADS Dong Y. , Yuen K. C. & Wu C. F. ( 2014a ) A multivariate regime-switching mean reverting process and its application to the valuation of credit risk. Stochastic Analysis and Applications , 32 , 687 – 710 . Google Scholar Crossref Search ADS Dong Y. H. , Yuen K. C. & Wu C. F. ( 2014b ) Unilateral counterparty risk valuation of CDS using a regime-switching intensity model. Statistics and Probability Letters , 85 , 25 – 35 . Google Scholar Crossref Search ADS Dong Y. , Yuen K. C. , Wang G. & Wu C. ( 2016 ) A reduced-form model for correlated defaults with regime-switching shot noise intensities. Methodology and Computing in Applied Probability , 18 , 459 – 486 . Google Scholar Crossref Search ADS Duffie D. & Gârleanu N. ( 2001 ) Risk and valuation of collateralized debt obligations. Financial Analysts Journal , 57 , 41 – 59 . Google Scholar Crossref Search ADS Duffie D. & Singleton K. ( 1999 ) Modeling Term Structures of Defaultable Bonds. Review of Financial Studies , 12 , 687 – 720 . Google Scholar Crossref Search ADS Elliott R. J. , Aggoun L. & Moore J. B. ( 1994 ) Hidden Markov Models: Estimation and Control . Berlin-Heidelberg-New York : Springer-Verlag . Gaspar R. M. & Schmidt T. ( 2010 ) Credit risk modeling with shot-noise processes , working paper. Social Science Electronic Publishing . Ge L. , Qiao X. S. & Yue X. Y. ( 2015 ) Explicit formulas for pricing credit-linked notes with counterparty risk under reduced-form framework. IMA Journal of Management Mathematics , 26 , 325 – 344 . Google Scholar Crossref Search ADS Giesecke K. ( 2003 ) A simple exponential model for dependent defaults. Journal of Fixed Income , 13 , 74 – 83 . Google Scholar Crossref Search ADS Giesecke K. , Longstaff F. A. , Schaefer S. & Ilya Strebulaev I. ( 2011 ) Corporate bond default risk: A 150-year perspective. Journal of Financial Economics , 102 , 233 – 250 . Google Scholar Crossref Search ADS Hamilton J. ( 1989 ) A new approach to the economic analysis of nonstationary time series. Econometrica , 57 , 357 – 384 . Google Scholar Crossref Search ADS Hull J. & White A. ( 2004 ) Valuation of a CDO and a nth to default CDS without Monte Carlo simulation. Journal of Derivatives , 12 , 8 – 23 . Google Scholar Crossref Search ADS Jarrow R. & Turnbull T. ( 1995 ) Pricing derivatives on financial securities subject to credit risk. Journal of Finance , 50 , 53 – 85 . Google Scholar Crossref Search ADS Leung K. S. & Kwok Y. K. ( 2009 ) Counterparty risk for credit default swaps: Markov chain interacting intensities model with stochastic intensity. Asia-Pacific Financial Markets , 16 , 169 – 181 . Google Scholar Crossref Search ADS Lindskog F. & McNeil A. ( 2003 ) Common Poisson shock models: applications to insurance and credit risk modelling. Astin Bulletin , 33 , 209 – 238 . Google Scholar Crossref Search ADS Lu Y. , Li S. ( 2009 ) The Markovian regime-switching risk model with a threshold dividend strategy. Insurance: Mathematics and Economics , 44 , 296 – 303 . Merton R. C. ( 1974 ) On the pricing of corporate debt: The risky structure of interest rates, Journal of Finance , 29 , 449 – 470 . Schönbucher P. & Schubert D. ( 2001 ) Copula dependent default risk in intensity models . Working Paper. Department of Statistics, Bonn University . Siu T. K. ( 2010 ) Bond pricing under a Markovian regime-switching jump-augmented Vasicek model via stochastic flows. Applied Mathematics and Computation , 216 , 3184 – 3190 . Google Scholar Crossref Search ADS Zhang X. , Elliott R. J. & Siu T. K. ( 2012 ) A stochastic maximum principle for a Markov regime-switching jump-diffusion model and its application to finance. SIAM Journal on Control and Optimization , 50 , 964 – 990 . Google Scholar Crossref Search ADS © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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IMA Journal of Management MathematicsOxford University Press

Published: Sep 7, 2018

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